Compression Programming of Shape Memory Polymers Below the Glass Transition Temperature

ABSTRACT

Compression programming of a shape memory polymer without the requirement of added heat, wherein the programming occurs at a temperature below the glass transition of the shape memory polymer. The shape memory polymer can be either a thermoset or a thermoplastic shape memory polymer.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority under 35 U.S.C. §119(e) from U.S. Provisional Application Ser. No. 61/483,196, filed 6 May 2011, entitled “Biomimetic Self-Healing Composite” the contents of which are fully incorporated by reference herein. This application is related to copending U.S. utility application Ser. No. (to be assigned) entitled “Thermosetting Shape Memory Polymers with Ability to Perform Repeated Molecular Scale Healing” in the name of Guoqiang Li et al., the contents of which are fully incorporated by reference herein.

STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

This invention was made with government support under grant number CMMI 0946740 awarded by the National Science Foundation. The Government has certain rights in the invention.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to polymeric materials, more particularly to thermosetting polymers, and more particularly to methods for programming a thermoset shape memory polymer at ambient temperatures below the glass transition temperature of the thermoset.

2. Description of Related Art

Polymers:

Polymers are large molecules (macromolecules) composed of repeating structural sub-units. These sub-units are typically connected by covalent chemical bonds. The term polymer encompasses a large class of compounds comprising both natural and synthetic materials with a wide variety of properties. Because of the extraordinary range of properties of polymeric materials, they play essential and ubiquitous roles in everyday life. These roles range from familiar synthetic plastics and elastomers to natural biopolymers such as nucleic acids and proteins that are essential for life.

Plastics, Thermoset and Thermoplastic:

A plastic is any of a wide range of synthetic or semi-synthetic organic solids that are moldable. Plastics are typically organic polymers of high molecular mass, but they often contain other substances. There are two types of plastics: thermoplastic polymers and thermosetting polymers. Thermoplastics are the plastics that do not undergo chemical change in their composition when heated and can be molded again and again. Examples include polyethylene, polypropylene, polystyrene, polyvinyl chloride, and polytetrafluoroethylene (PTFE). Common thermoplastics range from 20,000 to 500,000 amu.

In contrast, thermosets are assumed to have an effectively infinite molecular weight. These chains are made up of many repeating molecular units, known as repeat units, derived from monomers; each polymer chain will have several thousand repeating units. Thermosets can take shape once; after they have solidified, they stay solid. Thus, in a thermosetting process, a chemical reaction occurs that is irreversible. In contrast to thermoplastic polymers (discussed below), once hardened a thermoset resin cannot be reheated and melted back to a liquid form.

Thermoplastic Polymers

A thermoplastic polymer, also known as a thermosoftening plastic, is a polymer that turns to a viscous liquid when heated and freezes to a rigid state when cooled sufficiently. Thermoplastic polymers differ from thermosetting polymers (e.g. phenolics, epoxies) in that they can be remelted and remolded.

Thermoplastics are elastic and flexible above a glass transition temperature (T_(g)) specific for each thermoplastic. Between the T_(g) and the higher melting temperature (T_(m)) some thermoplastics have crystalline regions alternating with amorphous regions in which the chains approximate random coils. The amorphous regions contribute elasticity and the crystalline regions contribute strength and rigidity. Above the T_(m) all crystalline structure disappears and the chains become randomly interdispersed. As the temperature increases above T_(m), viscosity gradually decreases without any distinct phase change.

Thermoplastics can go through melting/freezing cycles repeatedly and the fact that they can be reshaped upon reheating gives them their name. However, this very characteristic of reshapability also limits the applicability of thermoplastics for many industrial applications, because a thermoplastic material will begin to change shape upon being heated above its T_(g) and T_(m).

Thermosetting Polymers

According to an IUPAC-recommended definition, a thermosetting polymer is a prepolymer in a soft solid or viscous state that changes irreversibly into an infusible, insoluble polymer network by curing. Thermoset materials are usually liquid or malleable prior to curing and designed to be molded into their final form, or used as adhesives. Others are solids like that of the molding compound used in semiconductors and integrated circuits (IC).

Curing of thermosetting polymers may be done, e.g., through heat (generally above 200° C. (392° F.)), through a chemical reaction (two-part epoxy, for example), or irradiation such as electron beam processing. A cured thermosetting polymer is often called a thermoset. The curing process transforms the thermosetting resin into a plastic or rubber by a cross-linking process. Energy and/or catalysts are added that cause the molecular chains to react at chemically active sites (unsaturated or epoxy sites, for example), linking into a rigid, 3-D structure. The cross-linking process forms a molecule with a larger molecular weight, resulting in a material with a heightened melting point. During the curing reaction, the molecular weight increases to a point so that the melting point is higher than the surrounding ambient temperature, and the material solidifies.

However, uncontrolled heating of the material results in reaching the decomposition temperature before the melting point is obtained. Thermosets never melt. Therefore, a thermoset material cannot be melted and re-shaped after it is cured. A consequence of this is that thermosets generally cannot be recycled, except as filler material.

Thermoset materials are generally stronger than thermoplastic materials due to their three-dimensional network of bonds. Thermosets are also better suited for high-temperature applications (up to their decomposition temperature). However, thermosets are generally more brittle than thermoplastics. Because of their brittleness, thermosets are vulnerable to high strain rate loading such as impact damage. Because many lightweight structures use fiber reinforced thermoset composites, impact damage, if not healed properly and timely, may lead to catastrophic structural failure.

Smart Materials:

“Smart materials” or “designed materials” are materials that have one or more properties that can be significantly changed in a controlled fashion by external stimuli, such as stress, temperature, moisture, pH, electric or magnetic fields. For example, a shape memory polymer (SMP) is a material in which large deformation can be induced and recovered through energy (often thermal) changes or stress changes (pseudoelasticity). Shape memory polymers have varying visual characteristics depending on their formulation. Shape memory polymers may be epoxy-based, such as those used for auto body and outdoor equipment repair; cyanate-ester-based, which are used in space applications; and acrylate-based, which can be used in very cold temperature applications, such as for sensors that indicate whether perishable goods have warmed above a certain maximum temperature.

Temperature-responsive shape memory polymers are materials which undergo changes upon temperature change. There are also several other types of shape memory polymers that undergo change based on other than thermal energy. For example, pH-sensitive shape memory polymers are materials that change in volume when the pH of the surrounding medium changes. Photomechanical materials change shape under exposure to light.

The shape of temperature-responsive SMPs can be repeatedly changed by heating above their glass transition temperature (T_(g)). When heated, they become flexible and elastic, allowing for easy configuration.

Once they are cooled, they will maintain their new shape. However, the SMPs will return to their original shapes when they are reheated above their T_(g). An advantage of shape memory polymer resins is that they can be shaped and reshaped repeatedly without losing their material properties, and these resins can be used in fabricating shape memory composites.

Shape memory polymer composites are high-performance composites, formulated using fiber or fabric reinforcement and shape memory polymer resin as the matrix. Due to the shape memory polymer matrix, these composites have the ability to be easily manipulated into various configurations when they are heated above their glass transition temperatures and exhibit high strength and stiffness in their frozen or glassy state at temperatures lower than their glass transition. SMPs can also be reheated and reshaped repeatedly without losing their material properties.

Most SMPs are thermoplastics. However, a limited number of thermoset SMPs have been identified. The thermoset SMPs have a glass transition temperature above which the thermoset can be molded. However, as thermosets, they do not have a melting temperature, and after curing the polymer is set and can never be re-molded. If a thermoset SMP continues to be heated beyond its glass transition, it will never melt but will instead decompose when it reaches its decomposition temperature.

Shape memory polymers have become increasingly used due to their low cost, malleability, damage tolerance, and large ductility (Lendlein et al., 2005; Otsuka and Wayman, 1998; Nakayama, 1991). These advantages enable them to be active in various applications such as micro-biomedical components, aerospace deployable equipment and actuation devices (Tobushi et al., 1996; Liu et al., 2004; Yakacki et al., 2007).

Lately, confined shape recovery of shape memory polymers has been used for repeatedly sealing/closing structural-length scale impact damages (Li and John, 2008; Nji and Li, 2010a; and John and Li, 2010). A biomimetic two-step self-healing scheme, close-then-heal (CTH), has been proposed by Li and Nettles (2010) and Xu and Li (2010), and further detailed by Li and Uppu (2010), for healing structural-length scale damage autonomously, repeatedly, and molecularly. This concept has been validated by Nji and Li (2010b). It is envisioned that SMPs will be used in light-weight self-healing structures.

A thermally responsive shape memory polymer is not smart without programming. A common programming cycle starts with a deformation of the SMP at a temperature above the glass transition temperature (T_(g)). While maintaining the shape (strain) or stress, the temperature is lowered below T_(g). With the subsequent removal of the applied load, a temporary shape is created and fixed. This completes the typical three-step programming process. The original permanent shape can then be recovered upon reheating above T_(g), which is the thermal response aspect of a thermally responsive shape memory polymer.

The programming and shape recovery complete a thermomechanical cycle. However, for practical applications such as large structures, programming at very high temperature is not a trivial task because it is a lengthy, labor-intensive, and energy-consuming process. There is a need for alternative programming approaches. Various types of programming have been conducted on SMPs using the traditional heating-loading-cooling-unloading method. If the applied load is a tensile force or stretch, it is called tension or drawing programming; if the applied load is a compressive force or shrink, it is called compression programming. If either drawing or compression programming is conducted at temperatures below T_(g), it can be called cold-drawing programming or cold-compression programming.

Several theories have been developed to explain the thermomechanical profiles of SMPs. Earlier rheological models (Tobushi et al., 1997; Bhattacharyya and Tobushi, 2000) were capable of capturing the characteristic shape memory behavior of SMPs but with limited prediction capability due to the loss of the strain storage and release mechanisms. Later developments such as mesoscale model (Kafka, 2001; Kafka, 2008) and molecular dynamic simulation (Diani and Gall, 2007) propelled the understanding to a rather detailed level. Recently, the phenomenological approach (Morshedian et al., 2005; Gall et al., 2005; Liu et al., 2006; Yakacki et al., 2007; Chen and Lagoudas, 2008a; Chen and Lagoudas, 2008b; Qi et al., 2008; Xu and Li, 2010) emerges to be an effective tool to macroscopically investigate the thermomechanical mechanisms of SMPs. The work by Liu et al. (2006) is a typical example of these various phenomenological models, which proposed a continuum mixture of a frozen and an active phase controlled by a sole temperature dependent first-order phase transition concept for the thermally activated SMPs. Although arguably treating the SMPs as a special elastic problem without consideration of the time dependence, the model reasonably captures the essential shape memory responses to the temperature event. However, the involvement of nonphysical parameters such as volume fraction of the frozen phase and stored strain resulted in a controversial nonphysical interpretation of the glass transition process. In order to address such issues, Nguyen et al. (2008) presented a revolutionary concept which attributes the shape memory effects to structural and stress relaxation rather than the traditional phase transition hypothesis. They proposed that the dramatic change in the temperature dependence of the molecular chain mobility, which describes the ability of the polymer chain segments to rearrange locally to bring the macromolecular structure and stress response to equilibrium, underpins the thermally activated shape memory phenomena of SMPs. The fact that the structure relaxes instantaneously to equilibrium at temperatures above T_(g) but responds sluggishly at temperature below T_(g), suggests that cooling macroscopically freezes the structure into a non-equilibrium configuration below T_(g), and thus allows the material to retain a temporary shape. Reheating above T_(g) reduces the viscosity, restores mobility and allows the structure to relax to its equilibrium configuration, which leads to shape recovery.

It is noted that cold-drawing programming of thermoplastic SMPs has been conducted by several researchers. Lendlein and Kelch (2002) indicated that shape memory polymer (SMP) can be programmed by cold-drawing but did not give many details. Ping et al. (2005) investigated a thermoplastic poly(ε-caprolactone) (PCL) polyurethane for medical applications. In this polymer, PCL was the soft segment, which could be stretched (tensioned) to several hundred percent at room temperature (15-20° C. below the melting temperature of the PCL segment). They found that the cold-drawing programmed SMP had a good shape memory capability. Rabani et al. (2006) also investigated the shape memory functionality of two shape-memory polymers containing short aramid hard segments and poly(c-caprolactone) (PCL) soft segments with cold-drawing programming. As compared to the study by Ping et al. (2005), the hard segment was different but the same soft segment PCL was used. Wang et al. (2010) further studied the same SMP as Ping et al. (2005). They used FTIR to characterize the microstructure change during the cold-drawing programming and shape recovery. They found that in cold drawing programming, the amorphous PCL chains orient first at small extensions, whereas the hard segments and the crystalline PCL largely maintain their original state. When stretched further, the hard segments and the crystalline PCL chains start to align along the stretching direction and quickly reach a high degree of orientation; the hydrogen bonds between the urethane units along the stretching direction are weakened, and the PCL undergoes stress-induced disaggregation and recrystallization while maintaining its overall crystallinity. When the SMP recovers, the microstructure evolves by reversing the sequence of the microstructure change during programming. Zotzmann et al. (2010) emphasized that a requirement for materials suitable for cold-drawing programming is their ability to be deformed by cold-drawing. Based on their discussion, it seems that an SMP with an elongation at break as high as 20% is not suitable for cold-drawing programming.

BRIEF SUMMARY OF THE INVENTION

We have discovered that SMPs can gain the shape memory capability, creating a non-equilibrium configuration at temperatures below T_(g). We disclose a method for isothermal compression programming of a shape memory polymer, said method comprising: applying force to a shape memory polymer at a temperature less than the glass transition temperature of the shape memory polymer in a magnitude sufficient to produce a temporary shape deformation of the shape memory polymer. The shape memory polymer can be a thermoset or a thermoplastic shape memory polymer. The shape memory polymer can optionally be a closed-celled foam. In certain embodiments the applied force is a prestrain, and the prestrain is larger than the yielding strain of the shape memory polymer. In certain embodiments the applied force is a prestrain, and the prestrain is less than 30, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, or 51% strain. When the applied force is a prestrain, the prestrain can be at least 105%, 110%, 115%, 120%, 125%, 130%, 135%, 140%, 145%, 150%, 160%, 170%, 180%, 190%, 200%, 210%, 220%, 225%, 230%, 235%, 240%, 245%, 250%, 275%, 300%, 325%, 350%, 375%, 400%, 425%, 450%, 475%, 500%, 525%, 550%, 575%, 600%, 625%, or 650% of the yielding strain of the shape memory polymer, with a proviso that the prestrain is never more than a 100% strain. When the applied force is a prestrain, the prestrain can be can be at least 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 30, 35, 40, 45, 50 or 55%. In certain embodiments, a method for isothermal compression programming of a shape memory polymer further comprises a stress relaxation time of at least 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 30, 45, 60, 75, 90, 105, 120, 150, 180, 210, 240 or 260 min. Methods in accordance with the invention comprise various non-mutually exclusive combinations of the features set forth herein.

DEFINITIONS

“Decomposition Temperature (T_(D))” is the temperature at which chemical bonds are broken or violent oxidation or fire occurs.

“Fixed strain” is the difference between the prestrain and the springback. At the end of programming, there is a rebound or springback when the load is removed.

“Glass transition temperature (T₈)”: the temperature at which amorphous polymers undergo a transition from a rubbery, viscous amorphous liquid (T>T_(g)), to a brittle, glassy amorphous solid (T<T_(g)). This liquid-to-glass transition (or glass transition for short) is a reversible transition. The glass transition temperature T_(g), if one exists, is always lower than the melting temperature, T_(m), of the crystalline state of the material. An amorphous solid that exhibits a glass transition is called a glass. Supercooling a viscous liquid into the glass state is called vitrification. Despite the massive change in the physical properties of a material through its glass transition, the transition is not itself a phase transition; rather it is a phenomenon extending over a range of temperatures and is defined by one of several conventions. Several definitions of T_(g) have been endorsed as accepted scientific standards. Nevertheless, all such definitions are to some extent arbitrary, and they can yield different numeric results. The various definitions of T_(g) for a given substance typically agree within a few degrees Kelvin.

“Healing Temperature (T_(H))”: The healing temperature can be defined functionally as a preferred temperature above the melting temperature where thermoplastic molecules overcome intermolecular barriers and are able to gain mobility and to more effectively diffuse within a material.

“Melting point (T_(m))”: The term melting point, when applied to polymers, is not used to suggest a solid-liquid phase transition but a transition from a solid crystalline (or semi-crystalline) phase to a still solid but amorphous phase. The phenomenon is more properly called the crystalline melting temperature. Among synthetic polymers, crystalline melting is only discussed with regards to thermoplastics, as thermosetting polymers decompose at high temperatures rather than melt. Consequently, thermosets do not melt and thus have no T_(m).

“Prestrain” is the maximum strain applied during programming.

“Relaxation time” is the time elapsed during the stress relaxation process.

“Shape fixity” is similar to strain fixity, suggesting that a temporary shape is fixed.

“Shape fixity ratio” is the ratio of the strain after programming over the prestrain.

“Strain recovery” is the amount of strain that is recovered during shape recovery process.

“Stress relaxation” occurs when, after a material reaches a certain deformation, the stress continuously reduces while the strain remains constant.

“Yield strain” is the strain corresponding to yielding. In the stress-strain curve, the change of slope signals the start of yielding.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 An illustration of a four-step thermomechanical cycle in accordance with the present invention: Programming (Step 1-Step 3) and shape recovery (Step 4).

FIG. 2 DMA results as functions of temperature: (a) solid line-storage modulus (b) dashed line-loss modulus.

FIG. 3 Shape fixity results at temperature below T_(g) for specimens programmed at different prestrain levels (5%, 10%, and 30%).

FIG. 4 Strain-time response during the entire thermomechanical cycle for specimens programmed with (Panel a) 30% and (Panel b) 10% prestrain. The four steps for the specimen with 120 min of stress relaxation time during programming are also shown.

FIG. 5 The 3-D thermomechanical cycle in terms of stress-strain-time for different stress relaxation times with prestrain levels of 10% and 30%.

FIG. 6 An analogous decomposition scheme for the deformation gradient.

FIG. 7 A linear rheological illustration for stress response.

FIG. 8 Numerical simulation for samples with 30% prestrain (FIGS. 8 a) and 10% prestrain (FIG. 8 b) during the entire thermomechanical cycle. The four steps for the entire thermomechanical cycle for the specimen with 120 min of stress relaxation time during programming are also shown.

FIG. 9 Recovery strain as a function of temperature for different heating rates.

FIG. 10 Recovery strain as a function of temperature for different heating profiles.

FIG. 11 Thermomechanical cycle results for different programming temperatures: (FIG. 11 a) programming followed with immediate heating recovery, (FIG. 11 b) programming followed with cooling then heating recovery.

FIG. 12: Flowchart of the MATLAB program.

FIG. 13 Thermal response to stress-free cooling.

FIG. 14 Stress-strain response of the SMP at different temperatures.

FIG. 15 Stress-strain response of the SMP at different strain rates.

FIG. 16. Thermal response for a stress-free, constant heating rate (q=0.56° C./min) test.

FIG. 17 DMA results for the SMP based syntactic foam and the pure SMP

FIG. 18 XPS spectra of the pure SMP and the SMP based syntactic foam for (FIG. 18 a) the C 1s electron, and (b) the O 1s electron

FIG. 19 Strain-time response during the entire thermomechanical cycle for specimens programmed with (FIG. 19 a) 30% prestrain and (FIG. 19 b) 20% prestrain (the four steps shown in the figures are for the curve with 120 min of stress relaxation time.)

FIG. 20 Viscoelastic behavior of the foam by creep test at room temperature

FIG. 21 SEM observation of (a) pristine specimen and (b) specimen after 30% cold-compression programming

FIG. 22 Thermo-mechanical cycle in terms of (FIG. 22 a) stress-strain-time and (FIG. 22 b) stress-strain-temperature responses for different stress relaxation time with a pre-strain level of 30% and 20%

FIG. 23 Equivalent scheme for the SMP matrix

FIG. 24 An analogous decomposition scheme for the deformation gradient FIG. 25 An arbitrary nonlinear damage model with a linear equivalent FIG. 26 A linear rheological illustration for stress response

FIG. 27 Comparison of numerical simulations with experimental results for the full thermomechanical cycle (a) strain evolution with 30% prestrain, (b) strain evolution with 20% prestrain, and (c) thermomechanical cycle in terms of stress-strain-time response

FIG. 28 Comparison of numerical simulation with test results for a 2-D traditional thermomechanical cycle.

FIG. 29 Thermomechanical cycle results for specimen with different Φp.

FIG. 30 Thermomechanical cycle results for specimens with different w.

FIG. 31 Thermal response to stress-free natural cooling.

FIG. 32 Stress-strain response of the SMP based syntactic foam at various temperatures.

FIG. 33 Stress-strain response of the SMP based syntactic foam at different strain rates.

FIG. 34 Thermal response for stress-free constant-rate heating.

DETAILED DESCRIPTION OF THE INVENTION

Disclosed for the first time is a novel thermomechanical programming process for thermally activated SMPs, either thermoplastic or thermosetting SMPs. In accordance with the present invention, a non-equilibrium configuration can be created and maintained in shape memory polymers (SMPs) below T_(g). A new and effective approach is set forth herein which programs glass transition-activated SMPs directly at temperatures well below T_(g). The 1-D compression programming below T_(g) and free shape recovery were extensively investigated both experimentally (Example 1) and analytically (Example 2).

Example 3 applies the data and information from Example 1 to a shape memory polymer (SMP)-based self-healing syntactic foam, which was found to be capable of self-sealing structural scale damage repeatedly, efficiently, and almost autonomously.

In Example 4, a structural-relaxation constitutive model featuring damage-allowable thermoviscoplasticity was developed to predict the nonlinear shape memory behavior of the SMP based syntactic foam programmed at glassy temperatures. After validation by both 1-D (compression) and 2-D (compression in longitudinal direction and tension in transverse direction) tests, the constitutive model was used to evaluate the effects of several design parameters on the thermomechanical behavior of the SMP based syntactic foam. It is concluded that the model is a useful tool for designing and training this novel self-healing composite.

Thus, instead of the heating followed by cooling, the programming was conducted at a constant temperature which was well below the T_(g) of the SMP. In one embodiment, this invention comprises an approach to program thermoset or thermoplastic SMPs directly at temperatures well below T_(g), which effectively simplifies the shape fixing process. 1-D compression programming below T_(g) and free shape recovery of a thermoset SMP were experimentally investigated. Functional stability of the shape fixity under various environmental attacks was also experimentally evaluated.

A mechanism-based thermoviscoelastic-thermoviscoplastic constitutive model incorporating structural and stress relaxation was developed to predict the nonlinear shape memory behavior of the SMP trained below T_(g). Comparison between the prediction and the experiment showed good agreement. The structure dependence of the thermomechanical behavior of the SMP was further discussed through a parametric study per the validated constitutive model. This study validates that programming by cold-compression is a viable alternative for thermally responsive thermoset SMPs.

In accordance with the present invention, a thermosetting SMP was programmed by cold-compression. The elongation at break is about 4% for this thermosetting SMP at temperature below T_(g), which is not suitable for cold-drawing (tensioning) programming.

The thermomechanical behavior of the thermally responsive thermoset SMP with a unique programming process at glassy temperature has been studied both experimentally and theoretically. Among the results of this work are:

(1) The approach of cold-compression programming of a thermosetting shape memory polymer was tested and modeled. The test results show that this is an effective and efficient method which achieves very large and durable shape fixity, and has similar shape memory capability to specimens programmed by the more lengthy, labor-intensive, and energy-consuming approach currently used.

(2) The concept that the shape memory effect in nature is a transition between equilibrium and nonequilibrium configuration of the SMP structure can explain the shape memory mechanism of a thermoset SMP programmed by cold-compression.

(3) It was found that the prestrain level should be larger than the yielding strain of the SMP in order to fix a temporary shape at temperatures below T_(s).

(4) Longer stress relaxation time leads to larger shape fixity ratio. The upper bound of the shape fixity is determined by the difference between the prestrain and the spring-back, which is the ratio of the relaxed stress over the relaxed modulus.

(5) A finite deformation theory and mechanism based thermoviscoelastic constitutive model has been developed to study the thermomechanical behavior of the SMP programmed by cold-compression. Because the pseudo-plasticity and structure evolution are incorporated, the model reasonably captures the essential characteristics of the shape memory response. A fairly good agreement has been reached between the testing and modeling.

(6) The parametric simulation study reveals that the shape memory behavior is highly dependent on the heating profile. A faster heating rate shifts the onset of recovery to a higher temperature.

(7) The effect of heating history further corroborates that the shape recovery response is more a thermodynamic structure evolution than a steady state variable-determined phase transition. Beyond the glass transition temperature, even without further heating to a higher temperature, an adequate time period of soaking can still help achieve the full recovery.

(8) As long as the programming occurs in glassy state, the programming at a higher temperature followed with an immediate heating recovery leads to a higher shape fixity ratio and has slight effect on the strain recovery. The recovery of the SMP programmed at a higher temperature followed by a cooling process initiates at a lower temperature and progresses at a faster rate.

(9) It seems that the time-temperature equivalence principle holds for the shape memory behavior. Similar shape recovery ratio can be achieved at a higher temperature with a shorter time period of soaking or a longer time period of soaking at a lower temperature.

The programming of thermoset SMPs at glassy temperatures was successfully applied to a SMP-based, self-healing syntactic foam. A structure-evolving, damage-allowable thermoviscoplastic model has been developed, which reasonably captured the most essential shape memory response during this process. Results of this study included:

(1) Cold programming was effective and efficient for SMP-based self-healing syntactic foam. Considerable recoverability was achieved, although some damage in glass hollow microsphere inclusions was inevitable.

(2) A finite deformation, continuum constitutive model was developed to study the thermomechanical behavior of the SMP-based self-healing syntactic foam programmed at glassy temperature. As thermoviscoplasticity, structural relaxation and inclusion damage mechanism are considered in the model, the model plausibly captures the essential elements of the shape memory response. A fairly good agreement has been reached between the modeling results and the experimental results.

(3) The parametric simulation study revealed preferred embodiments for SMP-based syntactic foam: a high volume fraction of microsphere inclusions leads to a low recovery ratio, and a high wall thickness ratio of the glass microballoons leads to a larger recovery strain. Particular optimized configurations are achieved by adjusting and balancing these parameters.

The current model is based on closed-cell SMP based syntactic foam. Preferred embodiments of the invention comprise programming of closed-cell SMP foams, although open-cell foams may also be used.

EXAMPLES Example 1 Testing of Thermomechanical Behavior of Thermoset Shape Memory Polymer Programmed by Cold-Compression

In this example the SMP specimens were isothermally and uniaxially compressed to a certain strain level and then held for relaxation while strain was maintained. It was found that meaningful fixity ratios were achieved efficiently with an adequate prestrain and various relaxation time periods.

The stability of the fixed temporary shape was then verified under various environmental attacks such as water immersion and ultraviolet light exposure. Subsequent free shape recovery tests proved that the permanent shape was also recoverable upon heating, similar to the specimens programmed using the traditional approach.

Experimental Methods

Raw Materials, Curing, and Specimen Preparation

The shape memory polymer was a polystyrene-based thermoset SMP resin system with a T_(g) of 62° C. commercially sold by CRG Industries under the name of Vertex. A hardening agent distributed by the same company was added to the SMP resin. The mixture was blended for 10 min before it was poured into a 229×229×12 mm steel mold and placed into a vacuum chamber at 40 kPa for 20 min for removal of any air pockets introduced during the mixing process. The resin was then cured in an oven at 79° C. for 24 hours, followed by 6 hours at 107° C. After curing, the SMP panel was de-molded and cut into 30×30×12 mm block specimens for further testing.

Dynamic Mechanical Analysis

In order to determine the glass transition zone of the SMP, the dynamic mechanical analysis (DMA) test was conducted on a DMA 2980 tester from TA instruments per ASTM D 4092. A rectangular sheet with dimensions of 17.5×11.9×1.20 mm was placed into a DMA single cantilever clamping fixture. A small dynamic load at 1 Hz was applied to a platen and the temperature was ramped from room temperature to 120° C. at a rate of 3° C./min. The amplitude was set to be 15 μm.

Coefficient of Thermal Expansion

The linear thermal expansion coefficient was measured by using a linear variable differential transducer (LVDT, Cooper Instruments LDT 200 series) system to record the specimen surface displacement and a Yokagawa DC100 data acquisition system to collect the thermocouple measurement of the temperature change. The temperature was ramped from room temperature to 100° C. at an average heating rate of 0.56° C./min. After equilibration for 30 minutes, the sample was naturally cooled down to room temperature.

Programming by Isothermal Flat-Wise Uniaxial Compression Test

Specimens were programmed at a temperature well below the T_(g) of the SMP, instead of the typical lengthy programming process above T_(g). In this example room temperature (20° C.) was adopted for programming. The programming was conducted by a uniaxial compression test. Uniaxial flat-wise compression was performed with a MTS QTEST150 electromechanical frame outfitted with a moveable furnace (ATS heating chamber) per the ASTM C 365 standard at a displacement rate of 1.3 mm/min to the test prestrain level. Temperature control and monitoring were achieved through a thermocouple placed in the chamber near the SMP specimen. Stress-strain responses were generated for different prestrain levels and stress relaxation time.

In this study, three prestrain levels (5%, 10%, and 30%), corresponding to the elastic zone (5%) and post-yielding zone (10% and 30%), respectively, were selected. The stress relaxation time was determined at 0 min, 30 min, 120 min, and 260 min for the 5% prestrain level, and 0 min, 5 min, 15 min, 30 min, and 120 min for the 10% and 30% prestrain levels. At least three effective specimens were tested for each prestrain level and stress relaxation time. Based on the test results (1) the strain should be greater than the yielding strain; (2) the strain is preferably as high as about 40%, which starts to see significant strain hardening; (3) strain rate affects the shape fixity, i.e., for the same programming strain, the higher the strain rate, the lower the shape fixity. For example, tests using a strain rate of about 1,000/s for cold-compression programming showed reduced shape fixity, while shape memory capability was not affected, i.e., strain rate was reduced as compared to a lower strain rate such as 0.01/s.

Free Shape Recovery Test

Once the specimens were programmed, an unconstrained strain recovery test was then implemented, where the compressed SMP specimen was heated to T_(high=)79° C. at an average heating rate of q=0.82° C./min. The same LVDT system was used to track the movement of the specimen during heating.

The thermomechanical cycle including programming and shape recovery is schematically shown in FIG. 1. The programming comprises three steps at a glassy temperature—typically (but not necessarily) conducted at a fixed glassy temperature (room temperature was used in this study): compression to the designed prestrain (Step 1), stress relaxation (Step 2), and removal of loading (Step 3).

Depending on the relaxation time, the entire programming takes from minutes to a couple of hours, compared to prior heat-based programming methods, which require refined temperature control and typically over 10 hours of programming time (Li and Nettles, 2010; Li and Uppu, 2010). Step 4, shape recovery, is similar to what has been done in prior methods.

Environmental Conditioning Tests

The capability for the SMP to maintain its shape fixity has been well established for specimens programmed by the prior high-temperature programming approach. Prior to the present invention, however there was no information about the ability to achieve or the functional stability of SMP programmed at a temperature below T_(g) under various environmental attacks. The stability of the temporary shape of the SMP specimens programmed in accordance with the invention was investigated for water immersion, ultraviolet light (UV) exposure and a combination of these two conditions. For the water immersion test, one programmed specimen was immersed in a cup of drinking water. The water level was about 2.5 cm above the surface of the specimen. For the UV exposure test, one programmed specimen was put in the same plastic cup without water. A 300-Watt Mog Base UV lamp, which had a wavelength ranging from 280 to 340 nm (mixed UV-A and UV-B light), was placed about 30 cm away from the transparent plastic cup. For the combined water immersion and UV exposure test, one programmed specimen was immersed in the same transparent plastic cup containing the same amount of drinking water. At the same time, the specimen was exposed to the same UV source with the same intensity. The specimens were monitored regularly for up to 3 months in order to record any dimension changes. In the first two weeks, the dimension of the specimens was measured every day and after that, the dimension was recorded every week. After 3 months of environmental attacks, the specimens were recovered using the same procedure as the non-attacked specimens.

Experimental Results

DMA Test Results

The experimental results in FIG. 2 illustrate the storage modulus and loss modulus of the SMP as functions of temperature. The glass transition zone and T_(g) can be found from the storage modulus per ASTM D 4092. The intersection between the tangent at the inflection point and the extrapolated tangent at the glassy state defines the lower limit and the intersection between the tangent at the inflection point and the extrapolated tangent at the rubbery plateau defines the upper limit of the glass transition zone. The average value in between them defines the glass transition temperature T_(g)=67.78° C. The listed value of T_(g)=62° C. by the distributor was determined by differential scanning calorimetry (DSC), which was about 4° C. lower than their DMA results. Therefore, the T_(g) provided by the manufacturer is consistent with our test results.

Uniaxial Strain-Controlled Compression Programming

The strain evolution during the material programming process, including the first three steps of the entire thermomechanical cycle in FIG. 1, is presented in FIG. 3. It is seen that shape fixity highly depends on the prestrain levels.

SMP specimens programmed at a 5% prestrain level could not fix a temporary shape, regardless of the length of the stress relaxation time. Upon removal of the load, immediate full spring-back was observed. For specimens programmed at 30% prestrain, however, a reasonable amount of strain was preserved, even when the load was instantly removed (zero relaxation time). With zero stress relaxation time, the shape fixity was still about 73%. Therefore, the level of prestrain does affect programming at glassy temperatures.

As documented in a previous study (Li and Nettles, 2010), the uniaxial compression yielding strain of the same thermosetting SMP is about 7% at the same glassy temperature. A 5% prestrain falls in the elastic region of the SMP. Therefore, immediate full springback occurs regardless of the relaxation time held. At 30% prestrain, the SMP specimen already yields and thus is able to maintain a reasonable temporary fixed strain even without stress relaxation. Therefore, a post-yielding prestrain level determines the success of the programming at glassy temperature.

It can also be observed from FIG. 3 that, with 30% prestrain, a longer stress relaxation time in Step 2 tends to enhance the shape fixity ratio. As the relaxation time continuously increased, the shape fixity asymptotically approached an upper bound, which is equal to the difference between the prestrain and elastic spring-back (ratio of the relaxed stress over the relaxed modulus). Further increase in the relaxation time can hardly bring up any significant increase in the shape fixity ratio.

With 10% prestrain, which is about 3% higher than the yield strain, a tendency similar to 30% prestrain is observed. Therefore, as long as the prestrain is above the yield strain, a certain amount of shape fixity can be realized. Of course, as the prestrain increases, the shape fixity also increases. For example, at zero stress relaxation time, the shape fixity is about 62.5% for 10% prestrain level, which is lower than the corresponding shape fixity of 73% for 30% prestrain level. It is also observed that the shape fixity with 10% prestrain plateaus earlier than that with 30% prestrain as stress relaxation time increases, possibly due to less viscoelastic and viscoplastic deformation with lower prestrain level.

Environmental Conditioning Test

The environmental attack test detected no change in specimen dimensions for any environmental conditions during the tests. Free shape recovery test showed almost the same recovery ratio as those non-attacked specimens. Since the observation time was up to 3 months and the environment conditions covered the most common working conditions, the stability of the non-equilibrium configuration created by cold-compression programming should be well confirmed. Thus, the temporary shape of the thermosetting SMP programmed at temperature below T_(g) is stable.

Free Recovery Test

FIG. 19 shows the entire thermomechanical cycles, including the unconstrained strain recovery during the heating process (Step 4 in FIG. 1). From FIG. 4 (a), which is programmed by 30% prestrain, it is observed that initially the programmed specimen only shows a slight and gradual thermal expansion. As the temperature approaches T_(g), the influence of the entropy change dominates, leading to a rapid strain recovery. At temperatures well above T_(g), most of the prestrain has been released and the strain converges to a stabilized value.

It is interesting to note that a similar sigmoidal-type strain recovery path is shared by all the specimens with differing relaxation times during programming, indicating that the strain release mechanism is generally independent of the holding time during programming. With 10% prestrain (FIG. 4 (b)), the shape recovery follows a tendency similar to that with 30% prestrain. A noticeable difference exists in the shape recovery ratio. With the 10% prestrain, the shape recovery ratio is about 100%, regardless of the stress relaxation time during programming; with the 30% prestrain, there is a small amount of strain that cannot be recovered. A possible reason is that with the 30% prestrain, some damage may have been created within the SMP specimen, which cannot be recovered during free shape recovery.

Overall, the shape memory capability of the thermosetting SMP programmed by cold-compression is considerable. The approach of programming at a glassy temperature is much simpler and easier to implement, and exhibits a considerable shape memory capability.

The 3-D stress-strain-time behaviors for the entire thermomechanical cycle, which include the three-step cold-compression programming process and the one step heating recovery, are shown in FIG. 5, for both the 10% and 30% prestrain levels. An extremely nonlinear, time- and temperature-dependent behavior is revealed. In-depth understanding of this complex thermomechanical behavior is elucidated by comprehensive constitutive modeling, which is developed in the following example.

Example 2 Constitutive Modeling of Thermomechanical Behavior of Thermoset Shape Memory Polymer Programmed by Cold-Compression

A continuum finite deformation based thermoviscoelastic model was developed to further elucidate the finding obtained in Example 1. The concept presented by Nauyen et al. (2008) that the shape memory effect reflects the transition between equilibrium and nonequilibrium configuration of the SMP structure was adopted and extended to the isothermal shape fixity process below T_(g). The Narayanaswamy-Moynihan model (Narayanaswamy, 1971; Moynihan et al., 1976) was incorporated to represent the structure relaxation. Comparisons with experiments showed that the model could fairly well reproduce the general thermomechanical behavior of the thermoset SMP. Subsequent parametric studies were conducted to explore the shape memory responses to different stimuli and different programming temperatures per the validated constitutive model.

Constitutive Modeling

General Consideration

The molecular resistance to inelastic deformation for amorphous thermoset SMPs below the glass transition temperature (T_(g)) mainly originates from two sources: the intermolecular resistance to segmental rotation and the entropic resistance to molecular alignment (Boyce et al., 1989, 2001).

The four-step thermomechanical cycle shown in FIG. 5 can be analyzed as follows: It is assumed that the plastic flow does not commence until the stressed material completely overcomes the free energy barrier to the molecular chain mobility, a restriction imposed on molecular chain motion from neighboring chains. Following the initial yield, molecular alignment occurs and subsequently alters the configurational entropy of the material (Step 1). Since the plastic strain develops in a rate-dependent manner, the length of relaxation time physically indicates the degree of the nonequilibrium configuration (Step 2). A relaxed configuration is then obtained after elastically unloading to a stress free state (Step 3). Due to the high material viscosity and vanishing chain mobility at the glassy programming temperature, the nonequilibrium structure is prevented from relaxing to the equilibrium state during the observed time frame, resulting in a retained temporary shape at the end of Step 3. Upon heating above T_(g) the viscosity decreases and chain mobility increases. The thermodynamically favorable tendency of increasing entropy allows the material to restore its equilibrium configuration and thus achieve shape recovery (Step 4).

Based on this understanding, a mechanism-based constitutive model was developed by incorporating the nonlinear structural relaxation model into the continuum finite-deformation thermoviscoelastic theory. The aim of this effort was to establish a quantitative understanding of the shape memory behavior of the thermally responsive thermoset SMP programmed at temperatures below T_(g). To keep the model simple, several basic assumptions were made for purposes of the modeling:

1) The SMP system is assumed to be macroscopically isotropic and homogeneous. The stress field is assumed to be uniform.

2) Heat transfer in the material is not considered. The temperature is treated as uniform throughout the entire body.

3) The structural relaxation and inelastic behavior of the material is assumed to be solely dependent on the temperature, time and stress.

4) The material is assumed to undergo no damage during the thermomechanical cycle.

Deformation Response

As illustrated in FIG. 6, any arbitrary thermomechanical path can be considered as a transition of the material between an initial reference configuration of an undeformed and unheated continuum body denoted by Ω₀ and a spatial configuration Ω of the deformed body which may have also experienced a certain temperature change. It is assumed that the configuration Ω₀ is either in thermodynamic equilibrium in rubbery state or in a stress-free glassy configuration originated from mechanically unconstrained cooling from high temperature. A deformation gradient

$F = \frac{\partial x}{\partial X}$

represents the tangent of a general nonlinear mapping x=x(X(t),T(t),t) of a material point from Ω₀ to Ω. This deformation mapping is then considered to be a combination of a thermal deformation and a mechanical deformation, which can be separated through a multiplicative decomposition scheme (Lu and Pister, 1975; Lion, 1997):

F _(T) =F _(M) F _(T)  (1)

Here, F_(M) defines the mechanical deformation gradient; F_(T) defines the mapping path from Ω₀ to Ω_(T), an intermediate heated configuration. Because the material is assumed to be isotropic, the thermal deformation gradient can be expressed as

F _(T) =J _(T) ^(1/3) I  (2)

where J_(T)=det (F_(T)) is the determinant of the thermal deformation gradient, representing the volumetric thermal deformation.

To separate the elastic and viscous responses, we introduce a multiplicative split of the mechanical deformation gradient into elastic and viscous components (Sidoroff, 1974; Lion, 1997):

F _(M) =F _(e) F _(v)  (3)

Although a discrete spectrum of nonequilibrium processes F_(M) ^(i)=F_(e) ^(i)F_(v) ^(i) (i=1, . . . N) (Govindjee and Reese, 1997) would be more appropriate to describe the general behavior of the real solid materials, only single stress relaxation is considered in the following derivation for the sake of convenience. The viscous part of the velocity gradient is then defined as:

L _(v) ={dot over (F)} _(v) F _(v) ⁻¹ =D _(v) +W _(v)  (4)

where D_(v) is the symmetric part of L_(v), representing the plastic stretch of the velocity gradient and W_(v) is the asymmetric component, representing the plastic spin. By applying the polar decomposition, we can also split F_(e) into a stretch (V_(e)) and a rotation (R_(e)) as:

F _(e) =V _(e) R _(e)  (5)

Structural Relaxation Response

A fictive temperature T_(f) based approach firstly introduced by Tool (1946) has been proved to be extremely successful in supplying the information about the free volume or the structure in the formulation of the free energy density. The fictive temperature T_(f) is an internal variable to characterize the actual thermodynamic state during the glass transition, defined as the temperature at which the temporary nonequilibrium structure at T is in equilibrium (Nguyen et al., 2008). It was assumed that the rate change of the fictive temperature is proportional to its deviation from the actual temperature and the proportionality factor depends on both T and T_(f) (Narayanaswamy, 1971), as indicated in the evolution equation (Tool, 1946):

$\begin{matrix} {\frac{T_{f}}{t} = {{K\left( {T,T_{f}} \right)}\left( {T - T_{f}} \right)}} & (6) \end{matrix}$

The Narayanaswamy-Moynihan model (NMM), discussed in detail by Donth and Hempel (2002), is an improvement for this approach. Instead of postulating a simple exponential relaxation mechanism governed by a single relaxation time (Tool, 1946), the non-exponential structural relaxation behavior as well as the spectrum effect were studied. It is assumed that the whole thermal history T(t) starts from a thermodynamic equilibrium state where T(t₀)=T_(f)(t₀). And Tool's fictive temperature is defined by:

T _(f)(t)=T(t)−∫_(t) ₀ ^(t)φ(Δζ)dT(t)  (7)

The response function is chosen, according to Moynihan et al. (1976), in the manner of a Kohlrausch function (Kohlrausch, 1847), in which the value of β describes the non-exponential characteristic of the relaxation process:

φ=exp[−(Δζ)^(β)], 0<β≦1  (8)

The dimensionless material time difference Δζ is introduced to linearize the relaxation process:

$\begin{matrix} {{\Delta Ϛ} = {{{Ϛ(t)} - {Ϛ\left( t^{\prime} \right)}} = {\int_{t^{\prime}}^{t}{\frac{t}{\tau_{s}}.}}}} & (9) \end{matrix}$

where the structural relaxation time τ_(s), a macroscopic measurement of the molecular mobility of the polymer, accounts for the characteristic retardation time of the volume creep (Hempel et al., 1999; Nguyen et al., 2008). As presented earlier, the structural relaxation in terms of τ_(s) is controlled by both the actual temperature T and the fictive temperature T_(f). A Narayanaswamy mixing parameter x was introduced to weigh the individual influence (Narayanaswamy, 1971):

$\begin{matrix} {{\tau_{s} = {\tau_{0}{\exp \left\lbrack {{B\left( {T_{g} - T_{\infty}} \right)}^{2}\left( {\frac{x}{T - T_{\infty}} + \frac{1 - x}{T_{f} - T_{\infty}}} \right)} \right\rbrack}}},{0 < x \leq 1}} & (10) \end{matrix}$

It can be observed that the term of (1-x) describes the contribution of T_(f). Here, T_(g) is the glass transition temperature. T_(∞) denotes the Vogel temperature, defined as (T_(g)-50) (° C.). τ₀ corresponds to the reference relaxation time at T_(g). B is the local slope at T_(g) of the trace of time-temperature superposition shift factor in the global William-Landel-Ferry (WLF) equation (William et al., 1955).

After obtaining the evolution profile of T_(f), we can then evaluate the isobaric volumetric thermal deformation corresponding to a temperature change from T₀ to T (Narayanaswamy, 1971; Scherer, 1990; Nguyen et al., 2008):

J _(T)(T,T _(f))=1+α_(r)(T _(f) −T ₀)+α_(g)(T−T _(f))  (11)

where α_(r) and α_(g) represent the long-time volumetric thermal expansion coefficients of the material in the rubbery state and the short-time response in the glassy state, respectively.

Stress Response

The mechanical behavior of amorphous glassy polymers under various temperature conditions has been extensively studied by numerous researchers (Boyce et al., 1988a, b; Treloar, 1958; Boyce et al., 1989; Govindjee and Simo, 1991; Arruda and Boyce, 1993; Bergstrom et al., 1998; Miehe and Keck, 2000; Boyce et al., 2001; Qi and Boyce, 2005). Although other approaches can still accommodate the present constitutive framework, the method of Boyce and co-workers was adopted in this study to model the general stress-strain behavior of the SMPs.

The overall mechanical resistance to the strain of a polymer mainly comes from two distinct sources: the temperature rat-dependent intermolecular resistance and the entropy-driven molecular network orientation resistance. It is possible to capture this nonlinear behavior by decomposing the stress response into an equilibrium time-dependent component σ_(ve) representing the viscoplastic behavior and an equilibrium time-independent component σ_(n) representing the rubber-like behavior. The two stress components can be represented by a three-element conceptual model as schematically illustrated in FIG. 7 for a one-dimensional analog. An elastic-viscoplastic component consists of an Eyring dashpot monitoring an isotropic resistance to chain segment rotation and a linear spring used to characterize the initial elastic response, while a parallel nonlinear hyperelastic element accounts for the orientation strain hardening behavior.

If we further denote the deformation gradient acting on the elastic-viscoplastic component by F_(ve) and the deformation gradient acting on the network orientation spring by F_(n), the following constitutive relations are revealed:

σ=σ_(ve)+σ_(n)  (12)

σ_(ve)=σ_(e)=σ_(v)  (13)

F _(ve) =F _(n) =F _(m)  (14)

F _(ve) =F _(e)F_(v)  (15)

The equilibrium response on the network orientation element can be defined following the Arruda-Boyce eight chain model (Arruda and Boyce, 1993) as:

$\begin{matrix} {\sigma_{n} = {{\frac{1}{J_{n}}\mu_{r}\frac{\lambda_{L}}{\lambda_{chain}}{\mathcal{L}^{- 1}\left( \frac{\lambda_{chain}}{\lambda_{L}} \right)}{\overset{\_}{B}}^{\prime}} + {{k_{b}\left( {J - 1} \right)}I}}} & (16) \end{matrix}$

where μ_(r) is the initial hardening modulus, and k_(b) denotes the bulk modulus to account for the incompressibility of rubbery behavior. Because most amorphous polymers exhibit vastly different volumetric and deviational behavior, the volumetric and deviational contributions are considered separately by taking out the volumetric strain through the split formulation (Flory, 1961; Simo et al., 1985):

F _(n) −J _(n) ^(−1/3) F _(n)  (17)

where J_(n)=det(F_(n)). B= F _(n) F _(n) ^(T), is the isochoric left Cauchy-Green tensor, and B′= B−⅓Ī_(n1)I represents the deviational component of B·Ī_(n1)=tr( B) is the first invariant of B. λ_(chain)=√{square root over (Ī_(n1)/3)} is the effective stretch on each chain in the eight-chain network. λ_(L) is the locking stretch representing the rigidity between entanglements. The Langevin function

is defined by:

$\begin{matrix} {{\mathcal{L}(\beta)} = {{\coth (\beta)} - \frac{1}{\beta}}} & (18) \end{matrix}$

whose inverse leads to the feature that the stress increases dramatically as the chain stretch approaches its limiting extensibility λ_(L).

The nonequilibrium stress response acting on the elastic-viscoplastic component can be determined through the elastic contribution F_(e):

$\begin{matrix} {\sigma_{ve} = {\sigma_{e} = {\frac{1}{J_{e}}{L^{e}\left( {\ln \; V_{e}} \right)}}}} & (19) \end{matrix}$

where J_(e)=det(F_(e)), and L^(e)=2G

+λI

I is the fourth order isotropic elasticity tensor. G and λ are Lamé constants,

is the fourth order identity tensor and I is the second order identity tensor.

The Viscous Flow

As proposed earlier, the molecular process of a viscous flow is to overcome the shear resistance of the material for local rearrangement. Therefore, a plastic shear strain rate {dot over (γ)}_(v) is given to help constitutively prescribe the viscous stretch rate D_(v) as:

D _(v)={dot over (γ)}_(v) n  (20)

where

$n = {\frac{\sigma_{ve}^{\prime}}{\sqrt{\sigma_{ve}^{\prime} \cdot \sigma_{ve}^{\prime}}} = \frac{\sigma_{ve}^{\prime}}{\sigma_{ve}^{\prime}}}$

is the normalized deviational portion of the nonequilibrium stress. This shows that the viscous stretch rate scales with the plastic shear strain rate and evolves in the direction of the flow stress.

Taking into account that the non-Newtonian fluid relationship must be valid for the dashpot of the mechanical model, the shear strain rate {dot over (γ)}_(v) can be formulated in an Eyring model (Eyring, 1936) with the temperature dependence in a WLF kinetics manner:

$\begin{matrix} {{\overset{.}{\gamma}}_{v} = {\frac{s}{\eta_{g}}\frac{T}{Q}{\exp \left( \frac{c_{1}\left( {T - T_{g}} \right)}{c_{2} + T - T_{g}} \right)}{\sinh \left( {\frac{Q}{T}\frac{\overset{\_}{\tau}}{s}} \right)}}} & (21) \end{matrix}$

here

$\overset{\_}{\tau} = \frac{\sigma_{ve}^{\prime}}{\sqrt{2}}$

is defined as the equivalent shear stress; c₁, c₂ are the two WLF constants; Q is the activation parameter; s represents the a thermal shear strength; and η_(g) denotes the reference shear viscosity at T_(g). The evolution Eq. (21) reveals the nature of the viscoplastic flow to be temperature-dependent and stress-activated.

More recently, Nguyen et al. (2008) further extended the viscous flow rule to a structure-dependent glass transition region by introducing the fictive temperature T_(j) into the temperature dependence:

$\begin{matrix} {{\overset{.}{\gamma}}_{v} = {\frac{s}{\eta_{g}}\frac{T}{Q}{\exp \left( {c_{1}\left( \frac{{c_{2}\left( {T - T_{f}} \right)} + {T\left( {T_{f} - T_{g}} \right)}}{T\left( {c_{2} + T - T_{g}} \right)} \right)} \right)}{\sinh \left( {\frac{Q}{T}\frac{\overset{\_}{\tau}}{s}} \right)}}} & (22) \end{matrix}$

It can be observed that once the material reaches equilibrium where T_(f)=T, Eq. (22) will reduce to Eq. (21) for a structure independent time-temperature shift factor.

Following yielding, the initial rearrangement of the chain segments alters the local structure configuration, resulting in a decrease in the shear resistance. To further account for the macroscopic post-yield strain softening behavior, the phenomenological evolution rule for athermal shear strength s proposed by Boyce et al. (1989) is implemented,

$\begin{matrix} {\overset{.}{s} = {{h\left( {1 - \frac{s}{s_{s}}} \right)}{\overset{.}{\gamma}}_{v}}} & (23) \end{matrix}$

The initial condition s=s₀ applies. Here s₀ denotes the initial shear strength, while s_(s) denotes the saturation value. h is the slope of the yield drop with respect to plastic strain. It should be noted that a softening characteristic can only be captured when s₀>s_(s) holds.

The constitutive relations for the sophisticated temperature- and time-dependent thermo-mechanical behavior of the thermally activated thermoset SMP are summarized in Table 1. The comprehensive model considers the material mechanical response in the manner of structure dependent thermoviscoelasticity. It is capable of capturing the important features of polymer behavior such as yielding, strain softening and strain hardening. Since our aim is to establish a thermomechanic framework for the extraordinary characteristics of SMPs programmed at glassy temperature, the present constitutive model does somewhat simplify real SMP behavior. Several factors such as heat conduction and pressure on the structure relaxation response are not taken into account. A single nonequilibrium stress relaxation process is also assumed for the sake of convenience, yet multiple relaxation mechanism (i.e., more separate Maxwell elements in FIG. 7) are required to distinguish the long-range entropic stiffening process and the short-range viscoplastic flow induced strain-hardening behavior.

Results

Model Validation

The constitutive relations were coded and implemented into a MATLAB program, for which a flowchart is illustrated in FIG. 12 to simulate the corresponding experimental data. The model parameters were obtained through various mechanical testing measurements. Detailed parameter identification procedures are briefly described below. The final values of these parameters are listed in Table 2. The mathematical formulation for 1-D compression is demonstrated below.

Based on the parameters in Table 2, the numerical simulation results, which cover the entire thermomechanical profile of the SMP programmed at 30% prestrain for different relaxation histories in a strain-time scope, is shown in FIG. 8 (a). The material was initially stressed to the pre-defined strain level after overcoming the yielding point and experiencing a slight strain-softening, followed by significant strain hardening (Step 1). Afterwards it was held with different time periods of relaxation for plastic strain development (Step 2). Finally the remaining stress was instantly removed, leading to a stress-free state (Step 3). Lengthy relaxation seemingly enhanced the level of the strain fixity. The stored deformation was then released and the original shape recovered during a subsequent heating process (Step 4).

TABLE 1 Summary of the thermoviscoelastic model deformation F = F_(e)F_(v)F_(T) response F_(T) = J_(T) ^(−1/3)I structure relaxation T_(f)(t) = T(t) − ∫_(t₀)^(t)ϕ(Δζ)dT(t) φ = exp[−(Δζ)^(β)] ${\Delta\zeta} = {{{\zeta (t)} - {\zeta \left( t^{\prime} \right)}} = {\int_{t^{\prime}}^{t}\frac{dt}{\tau_{s}}}}$ $\tau_{s} = {\tau_{0}\; {\exp \left\lbrack {{B\left( {T_{g} - T_{\infty}} \right)}^{2}\left( {\frac{x}{T - T_{\infty}} + \frac{1 - x}{T_{f} - T_{\infty}}} \right)} \right\rbrack}}$ stress σ = σ_(ve) + σ_(n) response $\sigma_{n} = {{\frac{1}{J_{n}}\mu_{T}\frac{\lambda_{L}}{\lambda_{chain}}{\mathcal{L}^{- 1}\left( \frac{\lambda_{chain}}{\lambda_{L}} \right)}{\overset{\_}{B}}^{\prime}} + {{k_{b}\left( {J - 1} \right)}I}}$ $\sigma_{ve} = {\frac{1}{J_{e}}{L^{e}\left( {\ln \; V_{e}} \right)}}$ viscous flow D_(v) = {dot over (γ)}_(v)n rule ${\overset{.}{\gamma}}_{v} = {\frac{s}{\eta_{g}}\frac{T}{Q}{\exp \left( {c_{1}\left( \frac{{c_{2}\left( {T - T_{f}} \right)} + {T\left( {T_{f} - T_{g}} \right)}}{T\left( {c_{2} + T - T_{g}} \right)} \right)} \right)}{\sinh \left( {\frac{Q}{T}\frac{\overset{\_}{\tau}}{s}} \right)}}$

TABLE 2 Material parameters of the preliminary constitutive model Model parameters Values T_(g) (° C.) glass transition temperature 62 T₀ (° C.) programming temperature 20 Δt (minute) relaxation time 0/5/15/30/120 α_(g) (10⁻⁴ ° C.⁻¹) volumetric CTE of glassy state 5.462 α_(r) (10⁻⁴ ° C.⁻¹) volumetric CTE of rubbery state 8.441 G (MPa) glassy shear modulus 196.4 λ (MPa) Lamé constant for glassy state 785.7 μ_(r) (MPa) rubbery modulus 1.2 k_(b) (MPa) bulk modulus 1000 λ_(L) locking stretch 0.95 μ_(g) (MPa · s⁻¹) reference shear viscosity at T_(g) 1550 s₀ (MPa) initial shear strength. 35 s_(s) (MPa) steady-state shear strength 33 Q/s₀ (° K/MPa) flow activation ratio 380 h (MPa) flow softening constant 250 c₁ first WLF constant 25.8 c₂ (° C.) second WLF constant 90 τ (s) structure relaxation characteristic time 200 x NMM constant 0.95 β Kohlrausch index 0.95

From FIG. 8 (a), the model simulation generally has a reasonable agreement with the test results. It proves that the model is capable of capturing the basic nonlinear material behavior of the SMP during a thermomechanical cycle. The real SMP samples did not achieve the full predicted recovery; this discrepancy may come from a couple of sources. Considering the large peak compressive stress applied during programming (about 40 MPa in FIG. 5 and FIG. 8( a)), some irreversible damage may have been induced in the SMP specimen. Also the deficiency of the single relaxation assumption appears evident in the discrepancies between the simulation and experiments when the relaxation time is insufficient. This can be validated by FIG. 8( a) that when the relaxation time is short, the discrepancy is large; when the relaxation time is long enough (120 min), the discrepancy becomes comparatively small. Therefore, a spectrum of multiple nonequilibrium processes would be required to describe the actual stress relaxation process of a real thermosetting SMP.

In this study, the same parameters calibrated in modeling the constitutive behavior of the SMP programmed by 30% prestrain level were also used to predict the thermomechanical behavior of the same SMP programmed by 10% prestrain level; see FIG. 8 (b). It is clear that, with the same set of parameters, the model predicted well the constitutive behavior of the SMP programmed by 10% prestrain. This further validated the developed model.

Prediction and Discussion

To demonstrate that the shape memory response of the SMP has a strong dependence on the structural evolution, the influence of the temperature profile has been investigated through the unconstrained recovery simulations.

Dependence on the Heating Rate

FIG. 9 exhibits the free recovery prediction results for two different heating rates q=0.6° C./min and q=3° C./min. It is observed that a faster heating rate shifts the initiation of the recovery process to a higher temperature and leads to a more gradual temperature dependence at the start of the strain release, but hardly affects the final recovery ratio.

Dependence on the Heating History

Besides the heating rate, the heating profile also influences the structure evolution. The calculation results for two types of heating profiles are shown in FIG. 10. Heating profile #1 represents a heating profile from 22° C. to 79° C. with a constant heating rate of q=1° C./min; while heating profile #2 represents a heating profile from 22° C. to 68° C. with a constant heating rate of q=1° C./min followed by a 50 minute soaking period. It shows that although heating profile #2 does not reach the same high temperature of 79° C. as that of the heating profile #1, it still reaches the same recovery strain level after adequate soaking. This is an indication of time-temperature equivalence.

Dependence on the Programming Temperature T₀

The effect of the programming temperature T₀ is shown in FIG. 11. The SMP samples are considered to be programmed at 20° C. and 40° C. respectively for the same relaxation time period of 20 minutes. Two cases are considered. For Case (a), shape recovery immediately follows the programming at a heating rate of 3° C./min, which means that the starting temperature for recovery is different (20° C. and 40° C., respectively). It can be seen that a higher T₀ significantly increases the shape fixity ratio due to the decrease of molecular segmental resistance during the plastic flow, and shortens the recovery time period. As the temperature-recovery strain subfigure in FIG. 11 (a) shows, the two programmed SMPs generally follow a similar recovery path except for the small deviation caused by the structure relaxation and thermal expansion. For Case (b), the sample programmed at 40° C. is first cooled to 20° C. before being heated to recover, which means that the starting temperature for recovery is the same (20° C.). It can be seen from FIG. 11 (b) that for the sample programmed at 40° C., it takes a longer time for completion of Step 4. The major recovery was completed at a lower temperature, again showing a time-temperature equivalency.

Detailed Parameter Identification Procedures

Although the final values of the material parameters used for demonstration, as listed in Table 2, were mainly obtained from curve fitting various testing results shown in FIG. 13 through FIG. 16, several basic guidelines were followed to assist in the estimates:

(1) A cooling profile of the thermal deformation is plotted versus the temperature in FIG. 13. The reference height L₀ denotes the initial sample height. It can be observed that the thermal response is not linear as the temperature traverses through the glass transition region. Linear α_(r) and α_(g) were computed from the slopes above and below the T_(g). Volumetric CTE is three times the value of the linear CTE.

(2) μ_(r) and λ_(L) are the parameters characterizing the rubbery behavior of the material, and can be determined from the stress-strain response at temperatures above T_(g) (FIG. 14). Lame constants G and A can be related to the initial slope of the isothermal uniaxial compression stress-strain curve in glassy state by assuming a typical polymer Poisson's ratio of 0.4 (Qi et al., 2008). Although it has been suggested that different sets of parameters μ_(r) and μ_(L) are preferred to capture the fundamentally different response of the rubbery state and the glassy state (Anand and Ames, 2006; Qi et al., 2008), they are treated as being temperature-independent for the sake of convenience in parameter identification and computational simplicity.

(3) As suggested in previous efforts (Boyce et al., 1989; Nguyen et al., 2008; Qi et al., 2008), the viscoplastic parameters such as Q, s, s, and h can be roughly determined from curve fitting of the compression tests at different strain rates (FIG. 15). The ratio Qls determines the strain rate dependence of the yield strength, and s/s_(s) indicates the drop of the shear strength. h characterizes the strain-softening rate after yielding.

(4) The structure relaxation parameters x and β are fitted to a stress-free, constant heating profile of the thermal deformation (FIG. 16).

Mathematical Formulation for 1-D Compression

For uniaxial compression, if we consider that the load is applied in the n₁ direction, the mathematical formula can be further reduced as follows:

Because of the assumption of isotropic material and uniform stress field,

$\begin{matrix} {F = \begin{bmatrix} \lambda_{1} & \; & \; \\ \; & \lambda_{2} & \; \\ \; & \; & \lambda_{2} \end{bmatrix}} & \left( {C{.1}} \right) \end{matrix}$

Here λ₁ represents the stretch in the n₁ direction and λ₂ is the stretch in the other two directions.

The isochoric left Cauchy strain tensor can be specified as:

$\begin{matrix} {{\overset{\_}{B} = {\left( J_{n} \right)^{{- 2}/3}\begin{bmatrix} \lambda_{1}^{2} & \; & \; \\ \; & \lambda_{2}^{2} & \; \\ \; & \; & \lambda_{2}^{2} \end{bmatrix}}},{J_{n} = {{\lambda_{1}\left( \lambda_{2} \right)}^{2}/J_{\tau}}}} & \left( {C{.2}} \right) \end{matrix}$

Hence the effective stretch λ_(chain) is defined as:

$\begin{matrix} {\lambda_{chain} = {\left( J_{n} \right)^{{- 1}/3}\sqrt{\frac{\lambda_{1}^{2} + {2\lambda_{2}^{2}}}{3}}}} & \left( {C{.3}} \right) \end{matrix}$

If λ₁ ^(e) and λ₂ ^(e) denote the elastic stretches, J_(e)=λ₁ ^(e)(λ₂ ^(e))² then the equilibrium and the non-equilibrium stresses can be identified by:

$\begin{matrix} {\sigma_{n} = {{\frac{\lambda_{1}^{2} - \lambda_{2}^{2}}{3J_{n}^{s/e}}\mu_{r}\frac{\lambda_{L}}{\lambda_{chain}}{{\mathcal{L}^{- 1}\left( \frac{\lambda_{chain}}{\lambda_{L}} \right)}\begin{bmatrix} 2 & \; & \; \\ \; & {- 1} & \; \\ \; & \mspace{11mu} & {- 1} \end{bmatrix}}} + {{k_{b}\left( {J - 1} \right)}I}}} & \left( {C{.4}} \right) \\ {\sigma_{ve} = {\frac{1}{J_{e}}\begin{bmatrix} {\ln \left( {\lambda_{1}^{e{({{2\; G} + \lambda})}}\lambda_{2}^{e\; 2\lambda}} \right)} & \; & \; \\ \; & {\ln \left( {\lambda_{1}^{e\; \lambda}\lambda_{2}^{e\; 2{({G + \lambda})}}} \right)} & \; \\ \; & \; & {\ln \left( {\lambda_{1}^{e\; \lambda}\lambda_{2}^{e\; 2{({G + \lambda})}}} \right)} \end{bmatrix}}} & \left( {C{.5}} \right) \end{matrix}$

$\overset{\_}{\tau} = \left. {\frac{2\sqrt{3}}{3J_{e}}G} \middle| {\ln \frac{\lambda_{2}^{e}}{\lambda_{2}^{e}}} \middle| . \right.$

As a result, the equivalent shear stress

Example 3 Testing of Shape Memory Polymer Based Self-Healing Syntactic Foam Programmed at Glassy Temperature

The novel process of programming at glassy temperatures has been set forth herein, and the recoverability and functional stability of thermosetting SMP programmed according to this “cold compression” programming method have been confirmed. In this example, the work is extended to SMP-based syntactic foams. Also, because of the composite nature and the damage tendency of the microballoons in the foam, a constitutive model underpinning the imperfect shape memory behavior developed and set forth in Example 4.

As set forth in Example 1, it was shown that, as long as a nonequilibrium configuration can be created for a glass-transition activated SMP, a temporary shape can be fixed, even if the temperature creating this nonequilibrium configuration is below the glass transition temperature. In other words, programming of SMPs can be conducted at glassy temperatures. A systematic experimental testing and constitutive modeling have validated this concept (also see [1]). We found that SMPs can be programmed at glassy temperature as long as the prestrain is greater than the yielding strain of the SMPs.

In this example, the three-step programming process set forth in Example 1 was applied to the SMP based syntactic foam at glassy temperatures. In laboratory testing the foam specimens were first programmed at glassy temperature with various stress relaxation time periods. Free shape recovery was then conducted. The shape fixity ratio and shape recovery ratio were determined. These test results were used as baseline data for the constitutive modeling set forth in Example 4.

Experimental Methods

Specimen Preparation

The SMP based syntactic foam was formulated through the dispersion of 40% by volume of glass hollow microspheres into the SMP matrix. The SMP named Veriflex from CRG Industries was used, a styrene-based thermoset SMP resin system (T_(g)=62° C.). The glass hollow microspheres were from Potters Industries (Q-CEL 6014) with an average outer diameter of 85 μm, an effective density of 0.14 g/cm³, and a wall thickness of 0.8 μm. The microspheres were incrementally added into the SMP resin, allowing several minutes for blending. A hardening agent was then added and the solution was blended for another 10 minutes before it was poured into a 229×229×12.7 mm steel mold. It was then placed in a vacuum chamber at 40 kPa for 20 minutes to remove any entrapped air bubbles. The curing process initiated at 79° C. for 24 hours, and then 107° C. for 3 hours, followed by 121° C. for 9 hours in an industrial oven, as recommended by Li and Nettles [7]. After curing, the foam panel was de-molded and was machined into different dimensions for various testing: 30×30×12.5 mm³ block specimens, which were determined per ASTM C365 standard [28], were used for thermal expansion, uniaxial compression, thermomechanical programming and shape recovery tests; and 17.5×11.9×1.20 mm³ plate specimens, which were determined per ASTM E1640-04 standard [29], were used for DMA tests. In this study, 40% by volume of microballoons was chosen for several reasons. (1) For most polymeric syntactic foams, the volume fraction of microballoons is around 40-60% [30]. (2) For this specific SMP, 40% was the volume fraction that maintained workability without the use of diluents. Diluents were not a preferred choice because they might affect the curing as well as the shape memory functionality of the foam. (3) This was the volume fraction we have used previously for the same foam [7]. Maintaining the same volume fraction facilitated comparisons.

Dynamic Mechanical Analysis

In order to determine the T_(g) of the foam, the single cantilever mode dynamic mechanical analysis (DMA) test was conducted on a DMA 2980 tester from TA instruments per ASTM E 1640-04 [29]. The specimen had a dimension of 17.5×11.9×1.20 mm³. The dynamic load frequency was set to be 1 Hz and the amplitude was 15 μm. The temperature ramped from room temperature to 120° C. at a rate of 3° C./min.

X-Ray Photoelectron Spectroscopy

The X-ray photoelectron spectroscopy (XPS) spectra of the pure SMP and the foam specimen were collected on a Kratos AXIS 165 high performance multi-technique surface analysis system with an information depth of 10 nm and a scan area of 700×300 μm². This was performed to qualitatively evaluate the interface between the SMP matrix and the glass hollow microspheres.

Thermal Expansion Measurement

A linear variable differential transducer (LVDT, Cooper Instruments LDT 200 series) system was used to measure the thermal expansion and a Yokagawa DC100 data acquisition system was used to monitor the temperature. The specimen was heated from room temperature to 100° C. at 0.4° C./min and naturally cooled down after thermally equilibrated for 30 minutes.

Programming of the Foam Below Glass Transition Temperature

The thermomechanical cycle including the new programming method and shape recovery was as schematically shown in FIG. 1. The programming comprised three steps at a fixed glassy temperature (e.g., room temperature in the present study): compression to the designed pre-strain (Step 1), stress relaxation (Step 2), and load removal (Step 3). Step 4 is the shape recovery step, which was conducted the same as in the traditional approach. Isothermal uniaxial flat-wise compression programming was performed on a MTS QTEST150 electromechanical frame outfitted with a moveable furnace (ATS heating chamber) per the ASTM C 365 standard [29]. The displacement rate was set to be 1.3 mm/min. A thermocouple placed in the chamber near the SMP specimen was used to control the environmental temperature.

As set forth in Example 1, successful shape fixity at glassy temperatures should have a post-yield pre-strain (i.e., a strain greater than yield strain). We tested prestrains below yielding strain, slightly above yielding strain, and well away from yielding but below fracture or significant strain hardening. Thus, two prestrain levels, 30% and 20%, which were above the yield strain of 7% for the same foam at room temperature [7], were selected with stress relaxation times of 0 min, 5 min, 15 min, 30 min, and 120 min. At least three effective specimens were tested for each stress relaxation time period.

Free Shape Recovery Tests

Unconstrained strain recovery tests were performed on the programmed specimens. During the test, the programmed foam specimen was reheated to T_(high)=80° C. at an average heating rate of q=0.4° C./min. The displacement at the specimen surface was tracked by the same LVDT system.

Experimental Results

DMA Test Results

The experimental results in FIG. 17 illustrate the loss modulus and storage modulus of the pure SMP and the SMP based syntactic foam as a function of temperature. It was found that the peak of the loss modulus of the foam had been shifted to a higher temperature as compared to that of the pure SMP. From FIG. 17, the difference in the T_(g) temperature was estimated to be 2.3° C. The T_(g) of 62° C. for the pure SMP provided by the manufacturer was determined by differential scanning calorimetry (DSC), which was about 6° C. lower than the DMA result from FIG. 17. To maintain consistency, we used the T_(g) of the pure SMP as 62° C. Therefore, the T_(g) of the foam was estimated to be 62° C.+2.3° C.=64.3° C.

XPS Test Results

The XPS results shown in FIG. 18 reveal that different binding energies exist in the pure SMP and the foam sample for the same emitted electrons (C (1s) and O (1s)). It indicated that some chemical shifts may have occurred at the glass hollow microsphere/SMP matrix interface. The mobility of the SMP polymer chains in the vicinity of the interface has probably been reduced, leading to an increase in glass transition temperature of the foam, which echoes the DMA test results.

Uniaxial Strain-Controlled Compression Programming

The strain evolution during the material programming process (Step 1-3) can be observed in FIG. 19. A reasonable shape fixity ratio (70.5% for 20% pre-strain and 72.6% for 30% pre-strain) was reached even when the constraint was instantly removed (zero relaxation time). Similar to the pure SMPs, it was found that longer stress relaxation times tend to increase the shape fixity ratio. However, an upper limit of the shape fixity ratio could be reached as the relaxation time continually increases. Further lengthening the relaxation time barely produced a noticeable increase in the shape fixity ratio.

The strain evolution with time (i.e., the change of strain with time) is further highlighted in FIG. 20 for viscoelastic tests. One is a creep test with a constant stress and the other with zero stress. It is clear that, even at room temperature, the foam showed creep. This is direct evidence that viscoelastic deformation can occur in the glassy state.

Therefore, a viscoelastic component was added in our modeling of Example 4. With zero stress, however, there is no change of strain with time, suggesting stability of the fixed level of strain.

Free Shape Recovery Test

FIG. 19 also shows the unconstrained heating recovery (Step 4). The programmed specimen initially showed slight thermal expansion. As the temperature further approached T_(g), the entropy increase led to a rapid strain recovery. At temperatures well above T_(g), the strain appeared to stabilize. A typical recovery path was shared by all the specimens with different relaxation times during programming, indicating a universal strain release mechanism. It was observed that the irrecoverable strain for all the specimens programmed by the same prestrain appeared to be at nearly the same level (about 8% for 20% pre-strain and 10% for 30% pre-strain), indicating a similar irrecoverable amount of damage occurred regardless of the relaxation time period. Therefore it is assumed that the damage occurred primarily in the compression process (Step 1). Since the damage in the SMP matrix itself under 30% prestrain can be neglected [1], the damage presumably came entirely from crushing and implosion of the glass hollow microspheres.

A Hitachi S-3600N VP-Scanning Electron Microscope was used to examine the microstructure change due to programming; see FIG. 21. From FIG. 21 (b), some of the microballoons have been crushed after cold-compression programming at 30% prestrain, which contributed to the irreversible strain after free shape recovery.

The extremely nonlinear behaviors for the entire thermomechanical cycle including a three-step glassy temperature programming process and a one-step heating recovery in the stress-strain-time view and the stress-strain-temperature view are shown in FIG. 22 (a) and FIG. 22 (b), respectively.

In-depth understanding of this complex thermomechanical behavior could be better elucidated by the constitutive modeling set forth in Example 4. It is noted that, as instant unloading occurs at the end of the programming, straight lines were used to connect the final loading point of Step 2 and the initial point of the free-recovery path in Step 4 in FIG. 22. These straight lines are not actual physical unloading curves, because the sudden removal of the load could not be recorded by the MTS machine. Therefore, the slopes of these straight lines do not represent the true unloading modulus.

Example 4 Thermoviscoplastic Modeling of Shape Memory Polymer Based Self-Healing Syntactic Foam Programmed at Glassy Temperature

As shown by the material characterization test results (DMA and XPS results), the incorporation of glass microballoons altered the chemical bonds at the interface between the SMP matrix and glass hollow microsphere inclusions. Earlier studies [31,32] reported that there exists a long-range gradient (over 100° K difference) for the polymer matrix glass transition temperature in the vicinity of the particles. Therefore, it was believed that an interfacial transition zone (ITZ) layer similar to the phenomenon in cement-based materials [33-35] also occurs in the SMP based syntactic foam. To consider the influence of such a layer on the performance of the foam, a unit cell of the SMP based syntactic foam was treated as a three-phase composite with ITZ-coated glass hollow microspheres embedded in the pure SMP matrix, as illustrated in FIG. 23. However, since current techniques have difficulties in characterizing the ITZ layer in details, a convenient approach of integrating the ITZ and pure SMP as a new equivalent SMP medium [25] was adopted. The equivalent scheme is also shown in FIG. 23 on the right.

Since the aim of this work was to establish a theoretical framework for the shape memory behavior of a damage-allowable SMP based syntactic foam programmed at glassy temperatures, several fundamental assumptions were made for further model derivation:

1) The material is considered to be isotropic, homogeneous and uniformly stressed.

2) The temperature is assumed to be spatially uniform.

3) Structural and stress relaxation are considered to be solely temperature, time and stress dependent.

4) The equivalent SMP matrix is considered to be thoroughly perfect. All the damage originates from the crushing and implosion of the glass hollow microspheres.

Kinematics

As documented previously, an arbitrary thermomechanical deformation mapping from an initial undeformed and unheated configuration Ω₀ to a spatial configuration Ω can be considered as a combination of a thermal deformation and a mechanical response; see FIG. 24. The scheme is expressed as a multiplicative decomposition of the deformation gradient [36,37]:

F=F _(M) F _(T) F=F _(M) F _(T)  (1)

where F_(M) defines the mechanical deformation gradient and F_(T) defines the mapping path from Ω₀ to Ω_(T), an intermediate heated configuration. Because the material is assumed to be macroscopically isotropic, the thermal deformation gradient is:

F _(T) =J _(T) ^(1/3) IF _(T) =J _(T) ^(1/3) I  (2)

where J_(T)=det(F_(T)) is the determinant of the thermal deformation gradient, representing the volumetric thermal deformation and I is the second order identity tensor.

To consider the composition of the syntactic foam, the rule of mixtures applies:

F _(M)=φ_(p) F _(p)(1−φ_(p))F _(i) F _(M)=φ_(p) F _(p)+(1−φ_(p))F _(i)  (3)

where F_(p) represents the deformation of the SMP matrix and F_(i) represents the deformation of the glass microsphere inclusions. Φ_(p) is the volume fraction of the polymer matrix.

Usually glass microspheres crush during the loading step; therefore, a damage allowable constitutive model of the microsphere inclusions is used. If an internal stress and time dependent evolution parameter Φ_(d)(σ,t) is introduced to represent the volume fraction of the damaged microspheres out of the total microsphere volume, the deformation of the inclusions could be expressed as:

F _(i)=(1−Φ_(d))F _(i) ^(ud)=Φ_(d) F _(i)=(1φ_(d))F _(i) ^(ud)+φ_(d) F _(i) ^(d)  (4)

where F_(i) ^(ud) refers to undamaged microspheres while F_(i) ^(d) refers to damaged microspheres.

To separate the elastic and viscous response of the SMP matrix, the multiplicative split scheme can be operated on the polymer deformation gradient [37,38]:

F _(p) =F _(p) ^(e) F _(p) ^(v) F _(p) =F _(p) ^(e) F _(p) ^(v)  (5)

where F_(p) ^(e) represents the elastic component and represents the viscous component.

Further polar decomposition of F_(p) ^(e) leads to a left stretch tensor and a rotation tensor

F _(p) =V _(p) ^(e) R _(p) ^(e)  (6)

The viscous velocity gradient is then defined as:

L _(p) ^(v) ={dot over (F)} _(p) ^(v) F _(p) ^(v-1) =D _(p) ^(v) W _(p) ^(v)  (7)

where D_(p) ^(v)=½(L _(p) ^(v) +L _(p) ^(eT)) represents the plastic stretch of the velocity gradient and is the spin. tensor.

4.3 Structural Relaxation and Thermal Deformation

The concept of fictive temperature T_(f) was first introduced by Tool [39] to explain the nonlinearity of structural relaxation. As defined, T_(f) is the temperature at which the temporary nonequilibrium structure at T is in equilibrium [26]. Considering that there exists an equilibrium configuration at a different temperature T_(f), which is equivalent to the current nonequilibrium configuration at the current temperature T, T_(f) serves as a measurement of the actual nonequilibrium structure state. The rate change of the fictive temperature is assumed to be proportionally dependent on its deviation from the actual temperature [40]. Its evolution was proposed as follows [39], where the temperature and structure dependent K represents the proportionality factor:

$\begin{matrix} {\frac{T_{f}}{t} = {{K\left( {T,T_{f}} \right)}\left( {T - T_{f}} \right)}} & (8) \end{matrix}$

The Narayanaswamy-Moynihan model (NMM) [40,41] further improved this approach by taking into account the non-exponential structural relaxation behavior as well as the spectrum effect. As discussed in detail by Donth and Hempel [42], with the assumption that the whole thermal history T(t) starts from a thermodynamic equilibrium state where T(t₀)=T_(f)(t₀), Tool's fictive temperature is given by:

T _(f)(t)=T(t)−∫_(t) ₀ ^(t)φ(Δζ)dT(t)T _(f)(t)=T(t)−∫_(t) ₀ ^(t)φ(Δζ)dT(t)  (9)

where φ is the response function and is expressed as a Kohlrausch function [43]:

φ=exp(−(Δζ)^(β))φ=exp[−(Δζ)^(β)], 0<β≦1  (10)

It is found from the equation above that, for very small departures from equilibrium is not constant [44]. Therefore β describes the non-exponential characteristic of the relaxation process.

Δζ is introduced as the dimensionless material time difference to linearize the relaxation process, roughly measuring the time in units of a mean structural relaxation time [45]:

$\begin{matrix} {{\Delta\varsigma} = {{{\varsigma (t)} - {\varsigma \left( t^{\prime} \right)}} = {{\int_{t^{\prime}}^{t}{\frac{t}{\tau_{s}}{\Delta\varsigma}}} = {{{\varsigma (t)} - {\varsigma \left( t^{\prime} \right)}} = {\int_{t^{\prime}}^{t}\frac{t}{\tau_{s}}}}}}} & (11) \end{matrix}$

where the parameter τ_(s), commonly referred to be the structural relaxation time, is a macroscopic measurement of the molecular mobility of the polymer [26,46]. As elaborated earlier that the structural relaxation is dependent on both T and T_(f), a Narayanaswamy parameter x was introduced to weigh their individual influence [40]:

$\begin{matrix} {{\tau_{s} = {{\tau_{0}{\exp \left\lbrack {{B\left( {T_{g} - T_{\infty}} \right)}^{2}\left( {\frac{x}{T - T_{\infty}} + \frac{1 - x}{T_{f} - T_{\infty}}} \right)} \right\rbrack}\tau_{s}} = {\tau_{0}{\exp \left\lbrack {{B\left( {T_{g} - T_{\propto}} \right)}^{2}\left( {\frac{x}{T - T_{\infty}} + \frac{1 - x}{T_{f} - T_{\infty}}} \right)} \right\rbrack}}}},,{0 < x \leq 1}} & (12) \end{matrix}$

It is understood that (1−x) describes the effect of the nonequilibrium state. T_(g) is the glass transition temperature and T_(∝)=T_(g)−50(° C.) denotes the Vogel temperature. T₀ corresponds to the reference relaxation time. B is the local slope at T_(g) of the trace of time-temperature superposition shift factor [47].

Since the material has been assumed to be statistically homogeneous and heat transfer is not considered, the global isobaric volumetric thermal deformation corresponding to a temperature change from T₀ to T can then be evaluated as follows [26,40,48]:

J _(T)(T,T _(f))=1+α_(r)(T _(f) −T ₀)+α_(g)(T−T _(f))  (13)

α_(r) and α_(g) respectively represent the long-term volumetric thermal expansion coefficients of the material in the rubbery state and the short-term response in the glassy state.

Constitutive Behavior of Glass Microsphere Inclusions

Since the glass hollow microspheres are brittle and have a high Young's modulus, the constitutive behavior of the undamaged portion can be considered to be purely elastic:

σ=σ_(i) =L _(i) ^(e)(ln F ^(ud))  (14)

where L_(i) ^(e)=2G_(i)

+λ_(i)I

I is the fourth order isotropic elasticity tensor of the glass microspheres. G_(i) and λ_(i) are Lamé constants, is the fourth order identity tensor and I is the second order identity tensor.

Physically, the evolution of the crushing and implosion of the hollow microspheres can be extremely complex. Since our focus is just on establishing a thermomechanical framework for the SMP based syntactic foam, for simplicity we assume an instant and complete damage mechanism occurring to the hollow microspheres partly because the glass hollow microspheres are brittle and thus the crack propagation speed is high. So Φ_(d)(σ,t)=Φ_(d)(σ).

Φ_(d)(σ) normally evolves nonlinearly. If a normal statistical distribution applies, then an arbitrary nonlinear curve of the volume fraction of the damaged microballoons should start slowly when the applied load initially overcomes the bearing stress σ_(b) and then should accelerate as the load further increases, and finally slow down gradually as damage proceeds and reaches a complete failure of all the microsphere inclusions, as illustrated in FIG. 25. Since it is difficult to capture the actual nonlinear damage profile, a linear equivalent damage model was considered. As the irrecoverable strain is assumed to fully come from the damage and volume reduction of the hollow microspheres, the total damage volume fraction (Φ_(d) ^(total)) of the microspheres can be calculated based on its relation to the final irrecoverable strain (ε_(ir)) as: 1+ε_(ir)=Φ_(p)+(1−φ_(p))((1−Φ_(d) ^(total))+Φ_(d) ^(total)(1+(1−w)³)^(1/8)), where w is the wall thickness ratio for the glass hollow microspheres. The proportionate factor k for the linear equivalent damage model is given by

${k = \frac{\varphi_{d}^{total}}{\left( {\sigma_{m} - \sigma_{b}} \right)}},$

where σ_(m) is the maximum stress during the programming process. Because the maximum stress is achieved at the end of loading in Step 1 of the programming process, the peak stress at the corresponding prestrain (30% or 20%) is used. σ_(b) corresponds to the initial damage stress, which is the crushing pressure of the glass microspheres as provided by the manufacturer (1.72 MPa). It is noted that the microballoons are not completely crushed (damaged) in the first programming cycle; see FIG. 21 (b). The damage should accumulate as the programming-recovery cycles increase and stabilize after several cycles, which may lead to a decrease in the shape recovery ratio in the first several cycles and an increase in the shape recovery ratio thereafter. For simplicity, however, the dependence of damage on the number of programming-recovery cycles was not considered in this study; this simplification could be a potential source of discrepancy between the model prediction and the test results.

If we additionally consider the glass microspheres to be isotropic, the damage gradient can be given by:

F _(d) =J _(d) ^(1/3)I  (15)

where J_(d) represents the ratio of the volume reduction during the damage, which can be determined as:

$\begin{matrix} {\mspace{79mu} {{I_{d} = {\frac{v_{a\; d}}{v_{b\; d}} = {\frac{\text{?}{\pi \left( {r^{B} - \left( {r - t} \right)^{B}} \right)}}{\text{?}\pi \; r^{B}} = {1 - \left( {1 - w} \right)^{3}}}}}{\text{?}\text{indicates text missing or illegible when filed}}}} & (16) \end{matrix}$

where V_(bd) and V_(ad) represent the volume of the hollow microsphere before and after damage, respectively; r is the outer radius of the microsphere; i is the wall thickness; and w=t/r is the wall thickness ratio.

It should be noted that even if completely crushed, the fractured pieces of the glass microspheres should still behave elastically. Hence, the deformation gradient of the damaged portion of the microspheres can be expressed as:

F _(i) ^(d) =F _(t) ^(ud) F _(d)  (17)

Constitutive Behavior of the Equivalent Shape Memory Polymer Matrix

Many efforts have been made to detail the constitutive relations of the highly nonlinear mechanical behavior of amorphous glassy polymers [49-59]. As the time-dependent mechanical behavior of the equivalent shape memory polymer involves equilibrium and nonequilibrium responses, a three-element conceptual model proposed by Boyce and co-workers, as illustrated in FIG. 26, were adopted to capture the stress response. A Maxwell element paralleling with a hyperelastic rubbery spring represents the stress split scheme:

σ=σ_(p)=σ_(p) ^(ve)σ_(p) ^(n)  (18)

here σ_(p) ^(ve) and are the stresses on the viscoplastic component and the rubbery spring.

The scheme indicates that the overall mechanical response to the straining can be expressed as the sum of the intermolecular segmental rotation resistance and the entropy driven molecular network orientation resistance. By further applying Hooke's Law to the linear elastic spring which characterizes the initial elastic response and Arruda-Boyce eight chain model [54] to the nonlinear rubbery spring which monitors the molecular network hardening, we can express the Cauchy stress as:

$\begin{matrix} {\sigma_{p} = {\left\lbrack {\frac{1}{J_{p}^{e}}{L_{p}^{e}\left( {\ln \; V_{p}^{e}} \right)}} \right\rbrack + \left\lbrack {{\frac{1}{J_{n}}\mu_{r}\frac{\lambda_{L}}{\lambda_{chain}}{L^{- 1}\left( \frac{\lambda_{chain}}{\lambda_{L}} \right)}\overset{\_}{B}} + {{k_{b}\left( {J_{n} - 1} \right)}I}} \right\rbrack}} & (19) \end{matrix}$

where the first part is and the second part is, J_(p) ^(e)=det(F_(p) ^(e)), and L_(p) ^(e) is the elasticity tensor; J_(n)=det(F_(p) ^(n)), and B=J_(n) ^(−2/3) F _(p) ^(n)F_(p) ^(nT) is the isochoric left Cauchy-Green tensor to consider the vastly different volumetric and deviational behavior exhibited by most amorphous polymers [60,61]; B′ is its deviational part; λ_(chain)=√{square root over (Ī_(n1)/3)} is the effective stretch; and Ī=tr( B) represents the first invariant. λ_(L) is the locking stretch representing the rigidity between entanglements. The Langevin function L is given by

${L(\beta)} = {{\coth (\beta)} - {\frac{1}{\beta}.}}$

The Eyring dashpot accounts for the isotropic resistance to the local molecular rearrangement such as chain rotation. A structure dependent viscous flow rule [26] was used to help describe its constitutive behavior:

$\begin{matrix} {{\overset{.}{\gamma}}_{v} = {\frac{s}{\eta_{g}}\frac{T}{Q}{\exp \left( {C_{1}\left( \frac{{c_{2}\left( {T - T_{f}} \right)} + {T\left( {T_{f} - T_{g}} \right)}}{T\left( {c_{2} + T - T_{g)}} \right.} \right)} \right)}{\sinh \left( {\frac{Q}{T}\frac{\overset{\_}{\tau}}{s}} \right)}}} & (20) \end{matrix}$

here

$\overset{\_}{\tau} = \frac{\sigma_{P}^{\prime \; {ve}}}{\sqrt{2}}$

is the equivalent shear stress; c₁, c₂ are WLF constants; Q is the activation parameter; η_(g) denotes the shear viscosity at T_(g); s represents the a thermal shear strength, and a phenomenological evolution rule

$\overset{.}{s} = {{h\left( {1 - \frac{s}{s_{S}}} \right)}{{\overset{.}{\gamma}}_{v}\left( {{s = s_{0}},{t = t_{0}}} \right)}}$

proposed by Boyce et al. [52] can be adopted to further feature the post-yield strain softening, where s₀ denotes the initial shear strength, s_(s) is the saturation value, and h describes the yield drop with respect to plastic strain; and {dot over (γ)}_(v) is the plastic shear strain rate. It is related to the viscous stretch rate D_(p) ^(v) as

${{\overset{.}{\gamma}}_{v} = {\frac{\sigma_{P}^{\prime \; {ve}}}{\sigma_{P}^{\prime \; {ve}}} = D_{P}^{v}}},$

indicating that the viscous stretch rate scales with the plastic shear strain rate and evolves in the direction of the flow stress. It is also noted that Eq. (20) will be reduced to the standard Eyring equation [62] upon thermal equilibrium where T_(f)=T.

Model Summary

The temperature- and time-dependent, damage-allowable thermo-mechanical constitutive relations for the SMP based syntactic foam are summarized in Table 3. The preliminary model considers the novel composite material in a structure-evolving manner. It was capable of capturing the essential mechanical behavior such as yielding, strain softening and strain hardening. The influence of the crushing and implosion of the glass hollow microspheres is also taken into account. However, since the focus is on developing a theoretical thermo-mechanical framework for the SMP based syntactic foam programmed at glassy temperature, the proposed constitutive model is rough as compared to the actual material behavior. Factors such as heat conduction, deformation-induced entropy change and pressure effects on the structure relaxation are excluded. A comparatively simple instant and complete-damage process is also assumed for the glass hollow microspheres. Detailed modeling efforts on the interaction between the matrix and inclusions would help capture the more vivid physical phenomenon.

TABLE 3 Summary of the constitutive model deformation F = F_(M)F_(T); F_(M) = φ_(p)F_(p) + (1 − φ_(p))F_(i) response F_(i) = (1 − φ_(d))F_(i) ^(ud) + φ_(d)F_(i) ^(d); F_(i) ^(d) = F_(i) ^(ud)F_(d); F_(d) = J_(d) ^(1/3)I; J_(d) = 1 − (1 − w)³ F_(p) = F_(p) ^(e)F_(p) ^(v); F_(T) = J_(T) ^(1/3)I J_(T) = 1 + α_(r)(T_(f) − T₀) + α_(g)(T − T_(f)) structure T_(f)(t) = T(t) − ∫_(t₀)^(t)ϕ(Δζ)dT(t) relaxation ${\phi = {\exp \left( {- ({\Delta\zeta})^{\beta}} \right)}};{{\Delta\zeta} = {{{\zeta (t)} - {\zeta \left( t^{\prime} \right)}} = {\int_{t^{\prime}}^{t}\frac{dt}{\tau_{s}}}}}$ $\tau_{s} = {\tau_{0}\; {\exp \left\lbrack {{B\left( {T_{g} - T_{\infty}} \right)}^{2}\left( {\frac{x}{T - T_{\infty}} + \frac{1 - x}{T_{f} - T_{\infty}}} \right)} \right\rbrack}}$ stress response σ = σ_(i) = σ_(p) = σ_(p) ^(ve) + σ_(p) ^(n) σ_(i) = L_(i) ^(e)(lnF_(i) ^(ud)) $\begin{matrix} {\sigma_{p} = {{\frac{1}{J_{p}^{e}}{L_{p}^{e}\left( {\ln \; V_{p}^{e}} \right)}} +}} \\ \left\lbrack {{\frac{1}{J_{n}}\mu_{r}\frac{\lambda_{L}}{\lambda_{chain}}{L^{- 1}\left( \frac{\lambda_{chain}}{\lambda_{L}} \right)}{\overset{\_}{B}}^{\prime}} + {{k_{b}\left( {J_{n} - 1} \right)}I}} \right\rbrack \end{matrix}$ viscous flow $D_{p}^{v} = {{\overset{.}{\gamma}}_{v}\frac{\sigma_{p}^{{ve}^{\prime}}}{\sigma_{p}^{{ve}^{\prime}}}}$ ${\overset{.}{\gamma}}_{v} = {\frac{s}{\eta_{g}}\frac{T}{Q}{\exp \left( {c_{1}\left( \frac{{c_{2}\left( {T - T_{f}} \right)} + {T\left( {T_{f} - T_{g}} \right)}}{T\left( {c_{2} + T - T_{g}} \right)} \right)} \right)}{\sinh \left( {\frac{Q}{T}\frac{\overset{\_}{\tau}}{s}} \right)}}$

TABLE 4 Material parameters of the preliminary thermoviscoplastic constitutive model Model parameters Values T_(g) (° C.) glass transition temperature 64.3 T₀ (° C.) programming temperature 20 Δt (minute) relaxation time 0/5/15/30/120 Φ_(p) Φ_(p) volume fraction of SMP matrix 0.6 α_(g) (10⁻⁴ ° C.⁻¹) volumetric CTE of glassy state 5.062 α_(r) (10⁻⁴ ° C.⁻¹) volumetric CTE of rubbery state 6.841 G_(i) (GPa) Shear modulus of glass hollow microspheres 27.7 λ_(i) (GPa) Lamé constant for glass hollow microspheres 41.5 k (MPa⁻¹) damage rate for glass hollow microspheres 0.02 w wall thickness ratio for glass hollow microspheres 0.019 G_(P) (MPa) glassy shear modulus of SMP 96.4 λ_(P)(MPa) Lamé constant for glassy state of SMP 385.7 μ_(r) (MPa) rubbery modulus of SMP 0.3 k_(b) (MPa) bulk modulus of SMP 1000 λ_(L) locking stretch 1.4 η_(g) (MPa · s⁻¹) reference shear viscosity at T_(g) 4050 s₀ (MPa) initial shear strength. 20 s_(s) (MPa) steady-state shear strength 18 Q/s₀ (° K/MPa) flow activation ratio 800 h (MPa) flow softening constant 200 c₁ first WLF constant 17.3 c₂ (° C.) second WLF constant 70 τ (s) structure relaxation characteristic time 20 x NMM constant 0.95 β Kohlrausch index 0.95

Results

Model Validation

The structure-evolving, damage-allowable thermoviscoplastic constitutive model was computed in MATLAB. The corresponding model parameters were mainly obtained by curve-fitting various thermal and mechanical testing results. The mechanical and material parameter values are listed in Table 4.

The numerical simulation results shown in FIG. 27( a), (b), and (c) cover the full thermomechanical cycle of the SMP based syntactic foam programmed at room temperature with the pre-strain of both 20% and 30% in both strain-time scale and stress-strain-time scale. All of the five different relaxation histories (0 min, 5 min, 15 min, 30 min, and 120 min) for both pre-strains are included. The material was initially compressed to the pre-defined strain level, which was beyond the yielding point. A slight strain-softening behavior appears followed by the strain hardening (Step 1). After different periods of relaxation for viscoplastic strain development (Step 2), the remaining stress constraint is instantly removed, leading to an externally stress-free state (Step 3). The temporary shape is fixed and lengthy relaxation apparently promotes the strain fixity. During the subsequent heating recovery (Step 4), the stored deformation is released, although there is a considerable amount of irrecoverable strain due to the damage of the glass hollow microspheres.

The simulation generally showed a reasonable agreement with the experimental results and captured most of the essential nonlinear material behavior, although less agreement on final recovery strain was found for samples programmed to 20% pre-strain than those programmed to 30% pre-strain. This may be because under 20% pre-strain, damage in the microballoons was considerably less than that under 30% prestrain and was below the linear interpolation prediction. In other words, the linear damage evaluation assumption is more appropriate for heavily damaged microballoons than for slightly damaged counterparts. It is also noted that the approximate nature of the single relaxation assumption appears evident. When the relaxation time is insufficient, such as 0 minutes, the discrepancy is particularly apparent. As the relaxation time further increases, the discrepancy becomes comparatively less significant. Multiple non-equilibrium relaxation processes would be required to more closely describe an actual stress relaxation.

The thermomechanical cycle for a 2-D traditional programming process as reported by Li and Xu [27] was also compared. The cruciform specimen was initially subjected to a constant load of 54.3 N (168.3 kPa) vertically in compression and horizontally in tension at 79° C., after which the conventional training method was followed to achieve shape fixity (cooling to room temperature for about ten hours while holding the load, and then removing the load completely and instantly). After that it was reheated to 79° C. at a heating rate of 0.3° C./min and equilibrated for 30 minutes for free recovery. The simulation results in FIG. 28 show the strain evolution in the horizontal and vertical directions during the entire thermomechanical cycle. Again, good agreement was found between the testing and modeling results.

Prediction and Discussion

The effects of the material composition on the thermomechanical behavior were numerically investigated.

Volume fraction of the SMP matrix Error! Objects cannot be created from editing field codes. Φ_(p)

The thermo-mechanical cycle prediction results of two specimens with different volume fractions of SMP matrix (Φ_(p)=0.5 and Φ_(p)=0.6) experiencing 40-minute relaxation period are shown in FIG. 29. The recovery heating rate was 0.4° C./min.

It is found that less SMP appeared to slightly increase the shape fixity ratio, which seems anomalous. Further observation of the heating recovery revealed that the seeming enhancement in shape fixity originated from an increase in glass hollow microsphere damage. This is because the specimen with less SMP experienced greater irreversible strain, and the loss of recoverability was noticeably greater than the gain in the shape fixity. Therefore, it is believed that lower Φ_(p) should lead to more damage and a lower recovery ratio.

Wall Thickness Ratio w

Further consideration was given to the wall thickness ratio of the hollow glass microspheres. FIG. 30 shows the full thermomechanical cycle prediction for two specimens with different w. The corresponding variation in microsphere strength was assumed to be negligible.

The specimen with a higher w was found to be able to achieve a larger recovery ratio (lower permanent strain), as it contained fewer voids and hence suffered less damage during programming. It is also interesting to notice that the shape fixity seemed to be hardly affected by the variation in w, because the same crushing strength was assumed. Although the irreversible deformation of microballoons with lower w may tend to increase the shape fixity ratio, the reduction in the reversible viscous deformation in SMP counterbalanced that tendency.

The final values of the model parameters, as listed in Table 4, were mainly obtained from curve fitting various testing results shown in FIG. 31 through FIG. 34. Several basic guidelines were used to assist the initial estimations:

(1) A cooling history for the SMP based syntactic foam is plotted as thermal deformation versus the temperature in FIG. 31. L₀ denotes the initial reference sample height. Because the cooling rate is extremely slow, an average of 0.17° C./min, the thermal shrinkage can be perceived as the structural response. Linear CTEs α_(r) and a_(g) were computed from the slopes at temperatures above and below T_(g). Volumetric CTE is three times the values of the linear CTE.

(2) μ_(r) and λ_(L) characterize the rubbery behavior of the material, and can be determined from the stress-strain response at temperatures above T_(g). The initial slope of the isothermal uniaxial compression stress-strain curve in glassy state gives an estimate for the Lame constant if a typical polymer Poisson ratio of 0.4 is assumed [22]. The final values for all these polymer mechanical parameters are fitted against the stress-strain curves at various temperatures, as shown in FIG. 32.

(3) The viscoplastic parameters such as Q, s, s₅, and h can be roughly fitted from the compression tests at different strain rates (FIG. 33). The ratio Q/s determines the strain rate dependence of the yield strength, and s/s_(s) represents the shear strength drop. h characterizes the post-yield strain-softening rate. It is found that noticeable discrepancies appear between the modeling prediction and test results in FIG. 33, especially at large strain. It is believed that more detailed consideration of the interaction between matrix and inclusions and a more realistic anisotropic flow model could be able to achieve a better agreement.

(4) The structural relaxation parameters x and β are fitted to a stress-free, constant heating profile of the thermal deformation (FIG. 34).

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All documents, including patents or published applications, journal papers, and other documents either cited in this specification, or relied upon for priority, are fully incorporated by reference herein. In the event of an otherwise irreconcilable conflict, the present specification shall control. 

1. A method for compression programming of a shape memory polymer, said method comprising: (a) applying a compressive force to a shape memory polymer at a temperature less than the glass transition temperature of the shape memory polymer, to deform the shape of the shape memory polymer; and (b) releasing the compressive force, while retaining a temporary shape deformation of the shape memory polymer.
 2. The method of claim 1, wherein the shape memory polymer is a thermoset shape memory polymer.
 3. The method of claim 1, wherein the shape memory polymer is a thermoplastic shape memory polymer.
 4. The method of claim 1, wherein the shape memory polymer comprises a closed-cell foam.
 5. The method of claim 1, wherein the compressing force step applies a prestrain to the shape memory polymer, and wherein the prestrain is larger than the yielding strain of the shape memory polymer.
 6. The method of claim 5, wherein the prestrain is less than 51% strain.
 7. The method of claim 6 wherein, the prestrain is less than 46% strain.
 8. The method of claim 1, wherein the compressing force step applies a prestrain to the shape memory polymer, and wherein the prestrain is at least 110% of the yielding strain of the shape memory polymer, and wherein the prestrain is less than 100% strain.
 9. The method of claim 8, wherein the prestrain is at least 150% of the yielding strain of the shape memory polymer.
 10. The method of claim 1, wherein the compressing force step applies a prestrain, and the prestrain is at least 7%.
 11. The method of claim 10, wherein the prestrain is at least 10%.
 12. The method of claim 1, wherein said releasing step comprises a period of stress relaxation of at least 10 minutes.
 13. The method of claim 1, wherein said compressing force step has a strain rate in a range of 10⁻⁴/second to 10³/second.
 14. The method of claim 1, additionally comprising the step of heating the shape memory polymer above the glass transition temperature, whereby the shape memory polymer returns from the deformed shape to the shape memory polymer's memory shape.
 15. A method for compression programming of a shape memory polymer, said method comprising: applying prestrain force to a shape memory polymer at a temperature less than the glass transition temperature of the shape memory polymer, wherein the prestrain is greater than the yielding strain of the shape memory polymer, and wherein a temporary shape deformation of the shape memory polymer is obtained.
 16. The method of claim 15, additionally comprising a period of stress relaxation.
 17. The method of claim 16, wherein the period of stress relaxation is at least 10 minutes.
 18. The method of claim 15, wherein the force-applying step further has a strain rate of 10⁻⁴/s to 10³/s.
 19. The method of claim 15, additionally comprising the step of heating the shape memory polymer above the glass transition temperature, whereby the shape memory polymer returns from the deformed shape to the shape memory polymer's memory shape. 